商务与经济统统计 第八版 (安德森 著) 中信出版社 课后答案 453页

'课后答案网，用心为你服务！大学答案---中学答案---考研答案---考试答案最全最多的课后习题参考答案，尽在课后答案网（www.khdaw.com）！Khdaw团队一直秉承用心为大家服务的宗旨，以关注学生的学习生活为出发点，旨在为广大学生朋友的自主学习提供一个分享和交流的平台。爱校园（www.aixiaoyuan.com）课后答案网（www.khdaw.com）淘答案（www.taodaan.com） Chapter1DataandStatisticsLearningObjectives1.Obtainanappreciationforthebreadthofstatisticalapplicationsinbusinessandeconomics.2.Understandthemeaningofthetermselements,variables,andobservationsastheyareusedinstatistics.3.Obtainanunderstandingofthedifferencebetweenqualitative,quantitative,crossectionalandtimeseriesdata.4.Learnaboutthesourcesofdataforstatisticalanalysisbothinternalandexternaltothefirm.5.Beawareofhowerrorscanariseindata.6.Knowthemeaningofdescriptivestatisticsandstatisticalinference.7.Beabletodistinguishbetweenapopulationandasample.8.Understandtheroleasampleplaysinmakingstatisticalinferencesaboutthepopulation.2-1 Solutions:1.Statisticscanbereferredtoasnumericalfacts.Inabroadersense,statisticsisthefieldofstudydealingwiththecollection,analysis,presentationandinterpretationofdata.2.a.9b.4c.Countryandroomratearequalitativevariables;numberofroomsandtheoverallscorearequantitativevariables.d.Countryisnominal;roomrateisordinal;numberofroomsandoverallscoreareratio.3.a.Averagenumberofrooms=808/9=89.78orapproximately90roomsb.2of9arelocatedinEngland;approximately22%c.4of9havearoomrateof$$;approximately44%4.a.10b.Fortune500largestU.S.industrialcorporationsc.Averagerevenue=$142,275.9/10=$14,227.59milliond.Usingthesampleaverage,statisticalinferencewouldletusestimatetheaveragerevenueforthepopulationof500corporationsas$14,227.59million.5.a.3b.Industrycodeisqualitative;revenuesandprofitarequantitative.c.Averageprofit=10,652.1/10=$1065.21milliond.8of10hadaprofitover$100million;80%e.1of10hadanindustrycodeof3;10%6.Questionsa,c,anddarequantitative.Questionsbandearequalitative.7.a.Thedataarenumericandthevariableisqualitative.b.Nominal8.a.2,013b.Qualitativec.Percentagessincewehavequalitativedata2-2 d.(0.28)(2013)=563.64Musthavebeen563or564.9.a.Qualitativeb.30of71;42.3%10.a.Quantitative;ratiob.Qualitative;nominalc.Qualitative(Note:Rankisanumericlabelthatidentifiesthepositionofastudentintheclass.Rankdoesnotindicatehowmuchorhowmanyandisnotquantitative.);ordinald.Qualitative;nominale.Quantitative;ratio11.a.Quantitative;ratiob.Qualitative;ordinalc.Qualitative;ordinal(assumingemployeescanberankedbyclassification)d.Quantitative;ratioe.Qualitative;nominal12.a.ThepopulationisallvisitorscomingtothestateofHawaii.b.Sinceairlineflightscarrythevastmajorityofvisitorstothestate,theuseofquestionnairesforpassengersduringincomingflightsisagoodwaytoreachthispopulation.Thequestionnaireactuallyappearsonthebackofamandatoryplantsandanimalsdeclarationformthatpassengersmustcompleteduringtheincomingflight.Alargepercentageofpassengerscompletethevisitorinformationquestionnaire.c.Questions1and4providequantitativedataindicatingthenumberofvisitsandthenumberofdaysinHawaii.Questions2and3providequalitativedataindicatingthecategoriesofreasonforthetripandwherethevisitorplanstostay.13.a.Quantitativeb.Timeserieswith7observationsc.Numberofriverboatcasinos.d.Timeseriesshowsarapidincrease;anincreasewouldbeexpectedin1998,butitappearsthattherateofincreaseisslowing.14.a.4b.Allfourvariablesarequantitative.c.Timeseriesdatafor1993to1996.2-3 15.Crossectionaldata.Itisbasedonthe1996performancedatathatwasavailableApril1997.16.a.Wewouldliketoseedatafromproducttastetestsandtestmarketingtheproduct.b.Suchdatawouldbeobtainedfromspeciallydesignedstatisticalstudies.17.Internaldataonsalariesofotheremployeescanbeobtainedfromthepersonneldepartment.ExternaldatamightbeobtainedfromtheDepartmentofLabororindustryassociations.18.a.(48/120)100%=40%inthesamplediedfromsomeformofheartdisease.Thiscanbeusedasanestimateofthepercentageofallmales60orolderwhodieofheartdisease.b.Thedataoncauseofdeathisqualitative.19.a.AllsubscribersofBusinessWeekatthetimethe1996surveywasconducted.b.Quantitativec.Qualitative(yesorno)d.Crossectional-1996wasthetimeofthesurvey.e.Usingthesampleresults,wecouldinferorestimate59%ofthepopulationofsubscribershaveanannualincomeof$75,000ormoreand50%ofthepopulationofsubscribershaveanAmericanExpresscreditcard.20.a.56%ofmarketbelongedtoA.C.Nielsen$387,325istheaverageamountspentpercategoryb.3.73c.$387,32521.a.ThetwopopulationsarethepopulationofwomenwhosemotherstookthedrugDESduringpregnancyandthepopulationofwomenwhosemothersdidnottakethedrugDESduringpregnancy.b.Itwasasurvey.c.63/3.980=15.8womenoutofeach1000developedtissueabnormalities.d.Thearticlereported“twice”asmanyabnormalitiesinthewomenwhosemothershadtakenDESduringpregnancy.Thus,aroughestimatewouldbe15.8/2=7.9abnormalitiesper1000womenwhosemothershadnottakenDESduringpregnancy.e.Inmanysituations,diseaseoccurrencesarerareandaffectonlyasmallportionofthepopulation.Largesamplesareneededtocollectdataonareasonablenumberofcaseswherethediseaseexists.22.a.AlladultviewersreachedbytheDenver,Coloradotelevisionstation.b.Theviewerscontactedinthetelephonesurvey.c.Asample.Itwouldclearlybetoocostlyandtimeconsumingtotrytocontactallviewers.2-4 23.a.Percentoftelevisionsetsthatweretunedtoaparticulartelevisionshowand/ortotalviewingaudience.b.AlltelevisionsetsintheUnitedStateswhichareavailablefortheviewingaudience.Notethiswouldnotincludetelevisionsetsinstoredisplays.c.Aportionofthesetelevisionsets.Generally,individualhouseholdswouldbecontactedtodeterminewhichprogramswerebeingviewed.d.Thecancellationofprograms,theschedulingofprograms,andadvertisingcostrates.24.a.Thisisastatisticallycorrectdescriptivestatisticforthesample.b.Anincorrectgeneralizationsincethedatawasnotcollectedfortheentirepopulation.c.Anacceptablestatisticalinferencebasedontheuseoftheword“estimate.”d.Whilethisstatementistrueforthesample,itisnotajustifiableconclusionfortheentirepopulation.e.Thisstatementisnotstatisticallysupportable.Whileitistruefortheparticularsampleobserved,itisentirelypossibleandevenverylikelythatatleastsomestudentswillbeoutsidethe65to90rangeofgrades.Chapter2DescriptiveStatistics:TabularandGraphicalMethodsLearningObjectives1.Learnhowtoconstructandinterpretsummarizationproceduresforqualitativedatasuchas:frequencyandrelativefrequencydistributions,bargraphsandpiecharts.2.Learnhowtoconstructandinterprettabularsummarizationproceduresforquantitativedatasuchas:frequencyandrelativefrequencydistributions,cumulativefrequencyandcumulativerelativefrequencydistributions.3.Learnhowtoconstructadotplot,ahistogram,andanogiveasgraphicalsummariesofquantitativedata.4.Beabletouseandinterprettheexploratorydataanalysistechniqueofastem-and-leafdisplay.5.Learnhowtoconstructandinterpretcrosstabulationsandscatterdiagramsofbivariatedata.2-5 Solutions:1.ClassFrequencyRelativeFrequencyA6060/120=0.50B2424/120=0.20C3636/120=0.301201.002.a.1-(.22+.18+.40)=.20b..20(200)=40c/dClassFrequencyPercentFrequencyA.22(200)=4422B.18(200)=3618C.40(200)=8040D.20(200)=4020Total2001003.a.360°x58/120=174°b.360°x42/120=126°c.NoOpinion16.7%Yes48.3%No35%2-6 d.7060504030Frequency20100YesNoNoOpinionResponse4.a.Thedataarequalitative.b.PercentTVShowFrequencyFrequencyMillionaire2448Frasier1530ChicagoHope714Charmed48Total:501004-7 c.30252015Frequency1050MillionaireFrasierChicagoCharmedTVShowCharmed8%Chicago14%Millionaire48%Frasier30%d.Millionairehasthelargestmarketshare.Frasierissecond.5.a.MajorRelativeFrequencyPercentFrequencyManagement55/216=0.2525Accounting51/216=0.2424Finance28/216=0.1313Marketing82/216=0.3838Total1.001004-8 b.90807060y5040Frequenc3020100ManagementAccountingFinanceMarketingMajorc.PieChartManagementAccounting25%24%Finance13%Marketing38%6.a.BookFrequencyPercentFrequency7Habits1016.66Millionaire1626.67Motley915.00Dad1321.67WSJGuide610.004-9 Other610.00Total:60100.00TheErnst&YoungTaxGuide2000withafrequencyof3,InvestingforDummieswithafrequencyof2,andWhatColorisYourParachute?2000withafrequencyof1aregroupedinthe"Other"category.b.Therankorderfromfirsttofifthis:Millionaire,Dad,7Habits,Motley,andWSJGuide.c.ThepercentofsalesrepresentedbyTheMillionaireNextDoorandRichDad,PoorDadis48.33%.7.RatingFrequencyRelativeFrequencyOutstanding190.38VeryGood130.26Good100.20Average60.12Poor20.04501.00Managementshouldbepleasedwiththeseresults.64%oftheratingsareverygoodtooutstanding.84%oftheratingsaregoodorbetter.Comparingtheseratingswithpreviousresultswillshowwhetherornottherestaurantismakingimprovementsinitsratingsoffoodquality.8.a.PositionFrequencyRelativeFrequencyPitcher170.309Catcher40.0731stBase50.0912ndBase40.0733rdBase20.036Shortstop50.091LeftField60.109CenterField50.091RightField70.127551.000b.Pitchers(Almost31%)c.3rdBase(3-4%)d.RightField(Almost13%)e.Infielders(16or29.1%)toOutfielders(18or32.7%)9.a/b.StartingTimeFrequencyPercentFrequency7:003157:304208:004208:307359:00210201004-10 c.BarGraph876y54Frequenc32107:007:308:008:309:00StartingTimed.9:007:0010%15%7:3020%8:3035%8:0020%e.Themostpreferredstartingtimeis8:30a.m..Startingtimesof7:30and8:00a.m.arenext.10.a.Thedatarefertoqualitylevelsofpoor,fair,good,verygoodandexcellent.b.RatingFrequencyRelativeFrequencyPoor20.03Fair40.07Good120.20VeryGood240.40Excellent180.30601.004-11 c.BarGraph3025y2015Frequenc1050PoorFairGoodVeryGoodExcellentRatingPieChartGoodFair20%7%Poor3%ExcellentVeryGood30%40%d.Thecourseevaluationdataindicateahighqualitycourse.Themostcommonratingisverygoodwiththesecondmostcommonbeingexcellent.11.ClassFrequencyRelativeFrequencyPercentFrequency12-1420.0505.015-1780.20020.018-20110.27527.521-23100.25025.524-2690.22522.54-12 Total401.000100.012.ClassCumulativeFrequencyCumulativeRelativeFrequencylessthanorequalto1910.20lessthanorequalto2924.48lessthanorequalto3941.82lessthanorequalto4948.96lessthanorequalto59501.0013.18161412y108Frequenc642010-1920-2930-3940-4950-591.0.8.6.4.201020304050604-13 14.a.b/c.ClassFrequencyPercentFrequency6.0-7.94208.0-9.921010.0-11.984012.0-13.931514.0-15.93152010015.a/b.WaitingTimeFrequencyRelativeFrequency0-440.205-980.4010-1450.2515-1920.1020-2410.05Totals201.00c/d.WaitingTimeCumulativeFrequencyCumulativeRelativeFrequencyLessthanorequalto440.20Lessthanorequalto9120.60Lessthanorequalto14170.85Lessthanorequalto19190.95Lessthanorequalto24201.00e.12/20=0.6016.a.RelativePercentStockPrice($)FrequencyFrequencyFrequency10.00-19.99100.404020.00-29.9940.161630.00-39.9960.242440.00-49.9920.08850.00-59.9910.04460.00-69.9920.088Total251.001004-14 12108y6Frequenc42010.00-20.00-30.00-40.00-50.00-60.00-19.9929.9939.9949.9959.9969.99StockPriceManyofthesearelowpricedstockswiththegreatestfrequencyinthe$10.00to$19.99range.b.EarningsperRelativePercentShare($)FrequencyFrequencyFrequency-3.00to-2.0120.088-2.00to-1.0100.000-1.00to-0.0120.0880.00to0.9990.36361.00to1.9990.36362.00to2.9930.1212Total251.001004-15 10987654Frequency3210-3.00to-2.00to-1.00to0.00to1.00to2.00to-2.01-1.01-0.010.991.992.99EarningsperShareThemajorityofcompanieshadearningsinthe$0.00to$2.00range.Fourofthecompanieslostmoney.17.CallDurationFrequencyRelativeFrequency2-3.950.254-5.990.456-7.940.208-9.900.0010-11.920.10Totals201.00Histogram10987y65Frequenc432102.0-3.94.0-5.96.0-7.98.0-9.910.0-11.9CallDuration4-16 18.a.Lowestsalary:$93,000Highestsalary:$178,000b.SalaryRelativePercent($1000s)FrequencyFrequencyFrequency91-10540.088106-12050.1010121-135110.2222136-150180.3636151-16590.1818166-18030.066Total501.00100c.Proportion$135,000orless:20/50.d.Percentagemorethan$150,000:24%2018161412108Frequency642091-105106-120121-135136-150151-165166-180Salary($1000s)e.19.a/b.NumberFrequencyRelativeFrequency140-14920.10150-15970.35160-16930.15170-17960.30180-18910.05190-19910.05Totals201.004-17 c/d.NumberCumulativeFrequencyCumulativeRelativeFrequencyLessthanorequalto14920.10Lessthanorequalto15990.45Lessthanorequalto169120.60Lessthanorequalto179180.90Lessthanorequalto189190.95Lessthanorequalto199201.00e.2015Frequency10514016018020020.a.Thepercentageofpeople34orlessis20.0+5.7+9.6+13.6=48.9.b.Thepercentageofthepopulationthatisbetween25and54yearsoldinclusivelyis13.6+16.3+13.5=43.4c.Thepercentageofthepopulationover34yearsoldis16.3+13.5+8.7+12.6=51.1d.Thepercentagelessthan25yearsoldis20.0+5.7+9.6=35.3.Sothereare(.353)(275)=97.075millionpeoplelessthan25yearsold.e.Anestimateofthenumberofretiredpeopleis(.5)(.087)(275)+(.126)(275)=46.6125million.21.a/b.ComputerRelativeUsage(Hours)FrequencyFrequency0.0-2.950.103.0-5.9280.566.0-8.980.169.0-11.960.1212.0-14.930.06Total501.004-18 c.3025y2015Frequenc10500.0-2.93.0-5.96.0-8.99.0-11.912.0-14.9ComputerUsage(Hours)d.60504030Frequency201003691215ComputerUsage(Hours)e.Themajorityofthecomputerusersareinthe3to6hourrange.Usageissomewhatskewedtowardtherightwith3usersinthe12to15hourrange.4-19 22.5786458702255688023523.LeafUnit=0.1637557813489361004511324.LeafUnit=101161202130671422715516028170234-20 25.98910246611457889122457131214415126.LeafUnit=0.104789911292001355683494856714-21 27.41366750038960114457799700013445566678880113445778990227or413466750035896011446577997000134475566678880113448577899022974-22 28.a.058111334415678992233355268336779404785560b.2000P/EPercentForecastFrequencyFrequency5-926.710-14620.015-19620.020-24620.025-2926.730-3400.035-39413.340-4413.345-4926.750-5400.055-5900.060-6413.3Total30100.029.a.4-23 y12TotalA505xB11213C21012Total181230b.y12TotalA100.00.0100.0xB84.615.4100.0C16.783.3100.0c.y12A27.80.0xB61.116.7C11.183.3Total100.0100.0d.CategoryAvaluesforxarealwaysassociatedwithcategory1valuesfory.CategoryBvaluesforxareusuallyassociatedwithcategory1valuesfory.CategoryCvaluesforxareusuallyassociatedwithcategory2valuesfory.30.a.4-24 564024y8-8-24-40-40-30-20-10010203040xb.Thereisanegativerelationshipbetweenxandy;ydecreasesasxincreases.31.MealPrice($)QualityRating10-1920-2930-3940-49Good53.833.92.70.0VeryGood43.654.260.521.4Excellent2.611.936.878.6Total100.0100.0100.0100.0Asthemealpricegoesup,thepercentageofhighqualityratingsgoesup.Apositiverelationshipbetweenmealpriceandqualityisobserved.32.a.EPSRatingSales/Margins/ROE0-1920-3940-5960-7980-100TotalA189B145212C11237D3115E213Total44691336b.EPSRatingSales/Margins/ROE0-1920-3940-5960-7980-100TotalA11.1188.89100B8.3333.3341.6716.67100C14.2914.2928.5742.86100D60.0020.0020.00100E66.6733.33100HigherEPSratingsseemtobeassociatedwithhigherratingsonSales/Margins/ROE.Ofthosecompanieswithan"A"ratingonSales/Margins/ROE,88.89%ofthemhadanEPSRatingof80or4-25 higher.Ofthe8companieswitha"D"or"E"ratingonSales/Margins/ROE,only1hadanEPSratingabove60.33.a.IndustryGroupRelativeStrengthSales/Margins/ROEABCDETotalA12249B1523112C13217D11125E123Total411710436b/c.ThefrequencydistributionsfortheSales/Margins/ROEdataisintherightmostcolumnofthecrosstabulation.ThefrequencydistributionfortheIndustryGroupRelativeStrengthdataisinthebottomrowofthecrosstabulation.d.Oncethecrosstabulationiscomplete,theindividualfrequencydistributionsareavailableinthemargins.34.a.807060504030RelativePriceStrength20100020406080100120EPSRatingb.OnemightexpectstockswithhigherEPSratingstoshowgreaterrelativepricestrength.However,thescatterdiagramusingthisdatadoesnotsupportsucharelationship.Thescatterdiagramappearssimilartotheoneshowing"NoApparentRelationship"inFigure2.19.35.a.4-26 900.0800.0700.0600.0500.0400.0GamingRevenue300.0200.0100.00.00.0100.0200.0300.0400.0500.0600.0700.0800.0HotelRevenueb.Thereappearstobeapositiverelationshipbetweenhotelrevenueandgamingrevenue.Highervaluesofhotelrevenueareassociatedwithhighervaluesofgamingrevenue.36.a.VehicleFrequencyPercentFrequencyF-Series1734Silverado1224Taurus816Camry714Accord612Total50100b.ThetwotopsellingvehiclesaretheFordF-SeriesPickupandtheChevroletSilverado.4-27 Accord12%F-SeriesCamry34%14%Taurus16%Silverado24%c.37.a/b.IndustryFrequencyPercentFrequencyBeverage210Chemicals315Electronics630Food735Aerospace210Totals:201004-28 87654Frequency3210BeverageChemicalsElectronicsFoodAerospaceIndustryc.38.a.MovieFrequencyPercentFrequencyBlairWitchProject15936.0PhantomMenace8920.2Beloved8519.3PrimaryColors5712.9TrumanShow5111.6Total441100.0b.Truman12%ColorsWitch13%36%Beloved19%Phantom20%4-29 c.Thepercentofmailpertainingto1999coverstoriesis36.0+20.2=56.2%39.a-d.CumulativeRelativeCumulativeRelativeSalesFrequencyFrequencyFrequencyFrequency0-499130.65130.65500-99930.15160.801000-149900.00160.801500-199930.15190.952000-249910.05201.00Total201.00e.141210y86Frequenc4200-499500-9991000-14991500-19992000-2499Sales40.a.ClosingPriceFrequencyRelativeFrequency0-97/890.22510-197/8100.25020-297/850.12530-397/8110.27540-497/820.05050-597/820.05060-697/800.00070-797/810.025Totals401.0004-30 b.ClosingPriceCumulativeFrequencyCumulativeRelativeFrequencyLessthanorequalto97/890.225Lessthanorequalto197/8190.475Lessthanorequalto297/8240.600Lessthanorequalto397/8350.875Lessthanorequalto497/8370.925Lessthanorequalto597/8390.975Lessthanorequalto697/8390.975Lessthanorequalto797/8401.000c.1210y86Frequenc420515253545556575ClosingPriced.Over87%ofcommonstockstradeforlessthan$40ashareand60%tradeforlessthan$30pershare.41.a.RelativeExchangeFrequencyFrequencyAmerican30.15NewYork20.10OvertheCounter150.75201.00b.EarningsPerRelativeShareFrequencyFrequency0.00-0.1970.350.20-0.3970.350.40-0.5910.050.60-0.7930.154-31 0.80-0.9920.10201.00Seventypercentoftheshadowstockshaveearningspersharelessthan$0.40.ItlookslikelowEPSshouldbeexpectedforshadowstocks.Price-EarningRelativeRatioFrequencyFrequency0.00-9.930.1510.0-19.970.3520.0-29.940.2030.0-39.930.1540.0-49.920.1050.0-59.910.05201.00P-ERatiosvaryconsiderably,butthereisasignificantclusterinthe10-19.9range.42.RelativeIncome($)FrequencyFrequency18,000-21,999130.25522,000-25,999200.39226,000-29,999120.23530,000-33,99940.07834,000-37,99920.039Total511.000252015Frequency105018,000-21,99922,000-25,99926,000-29,99930,000-33,99934,000-37,999PerCapitaIncome4-32 43.a.0891022234441556666778888999201222344425683013b/c/d.NumberAnsweredRelativeCumulativeCorrectlyFrequencyFrequencyFrequency5-920.050210-1480.2001015-19150.3752520-2490.2253425-2930.0753730-3430.07540Totals401.000e.Relativelyfewofthestudents(25%)wereabletoanswer1/2ormoreofthequestionscorrectly.ThedataseemtosupporttheJointCouncilonEconomicEducation’sclaim.However,thedegreeofdifficultyofthequestionsneedstobetakenintoaccountbeforereachingafinalaconclusion.44.a/b.HighTemperatureLowTemperature339443685750002445579614444686187357972455801146890239c.Itisclearthattherangeoflowtemperaturesisbelowtherangeofhightemperatures.Lookingatthestem-and-leafdisplayssidebyside,itappearsthattherangeoflowtemperaturesisabout20degreesbelowtherangeofhightemperatures.d.Therearetwostemsshowinghightemperaturesof80degreesorhigher.Theyshow8citieswithhightemperaturesof80degreesorhigher.4-33 e.FrequencyTemperatureHighTemp.Low.Temp.30-390140-490350-5911060-697270-794480-895090-9930Total202045.a.8075e706560555045LowTemperatur403530405060708090100HighTemperatureb.Thereisclearlyapositiverelationshipbetweenhighandlowtemperatureforcities.Asonegoesupsodoestheother.46.a.SatisfactionScoreOccupation30-3940-4950-5960-6970-7980-89TotalCabinetmaker243110Lawyer1521110PhysicalTherapist521210SystemsAnalyst214310Total1710118340b.SatisfactionScoreOccupation30-3940-4950-5960-6970-7980-89Total4-34 Cabinetmaker20403010100Lawyer1050201010100PhysicalTherapist50201020100SystemsAnalyst20104030100c.Eachrowofthepercentcrosstabulationshowsapercentfrequencydistributionforanoccupation.Cabinetmakersseemtohavethehigherjobsatisfactionscoreswhilelawyersseemtohavethelowest.Fiftypercentofthephysicaltherapistshavemediocrescoresbuttherestareratherhigh.47.a.40,00035,00030,00025,00020,000Revenue$mil15,00010,0005,0000010,00020,00030,00040,00050,00060,00070,00080,00090,000100,000Employeesb.Thereappearstobeapositiverelationshipbetweennumberofemployeesandrevenue.Asthenumberofemployeesincreases,annualrevenueincreases.48.a.FuelTypeYearConstructedElecNat.GasOilPropaneOtherTotal1973orbefore4018312572471974-19792426220541980-19863738106821987-19914870201121Total14931717714504b.YearConstructedFrequencyFuelTypeFrequency1973orbefore247Electricity1491974-197954Nat.Gas3171980-198682Oil171987-1991121Propane7Total504Other14Total5044-35 c.CrosstabulationofColumnPercentagesFuelTypeYearConstructedElecNat.GasOilPropaneOther1973orbefore26.957.770.571.450.01974-197916.18.211.828.60.01980-198624.812.05.90.042.91987-199132.222.111.80.07.1Total100.0100.0100.0100.0100.0d.Crosstabulationofrowpercentages.FuelTypeYearConstructedElecNat.GasOilPropaneOtherTotal1973orbefore16.274.14.92.02.8100.01974-197944.548.13.73.70.0100.01980-198645.146.41.20.07.3100.01987-199139.757.81.70.00.8100.0e.ObservationsfromthecolumnpercentagescrosstabulationForthosebuildingsusingelectricity,thepercentagehasnotchangedgreatlyovertheyears.Forthebuildingsusingnaturalgas,themajoritywereconstructedin1973orbefore;thesecondlargestpercentagewasconstructedin1987-1991.Mostofthebuildingsusingoilwereconstructedin1973orbefore.Allofthebuildingsusingpropaneareolder.ObservationsfromtherowpercentagescrosstabulationMostofthebuildingsintheCG&Eserviceareauseelectricityornaturalgas.Intheperiod1973orbeforemostusednaturalgas.From1974-1986,itisfairlyevenlydividedbetweenelectricityandnaturalgas.Since1987almostallnewbuildingsareusingelectricityornaturalgaswithnaturalgasbeingtheclearleader.49.a.Crosstabulationforstockholder"sequityandprofit.Profits($000)Stockholders"Equity($000)0-200200-400400-600600-800800-10001000-1200Total0-12001011121200-24004102162400-3600433111133600-48001234800-60002316Total1816624450b.CrosstabulationofRowPercentages.Profits($000)Stockholders"Equity($1000s)0-200200-400400-600600-800800-10001000-1200Total4-36 0-120083.338.330.000.000.008.331001200-240025.0062.500.000.0012.500.001002400-360030.7723.0823.087.697.697.691003600-48000.000.000.0033.3366.671004800-60000.0033.3350.0016.670.000.00100c.Stockholder"sequityandprofitseemtoberelated.Asprofitgoesup,stockholder"sequitygoesup.Therelationship,however,isnotverystrong.50.a.Crosstabulationofmarketvalueandprofit.Profit($1000s)MarketValue($1000s)0-300300-600600-900900-1200Total0-8000234278000-1600044221216000-24000211424000-32000121432000-40000213Total27136450b.CrosstabulationofRowPercentages.Profit($1000s)MarketValue($1000s)0-300300-600600-900900-1200Total0-800085.1914.810.000.001008000-1600033.3333.3316.6716.6710016000-240000.0050.0025.0025.0010024000-320000.0025.0050.0025.0010032000-400000.0066.6733.330.00100c.ThereappearstobeapositiverelationshipbetweenProfitandMarketValue.Asprofitgoesup,MarketValuegoesup.51.a.ScatterdiagramofProfitvs.Stockholder"sEquity.4-37 1400.01200.01000.0800.0600.0Profit($1000s)400.0200.00.00.01000.02000.03000.04000.05000.06000.07000.0Stockholder"sEquity($1000s)b.ProfitandStockholder"sEquityappeartobepositivelyrelated.52.a.ScatterdiagramofMarketValueandStockholder"sEquity.4-38 45000.040000.035000.030000.025000.020000.015000.0MarketValue($1000s)10000.05000.00.00.01000.02000.03000.04000.05000.06000.07000.0Stockholder"sEquity($1000s)b.ThereisapositiverelationshipbetweenMarketValueandStockholder"sEquity.Chapter3DescriptiveStatistics:NumericalMethodsLearningObjectives1.Understandthepurposeofmeasuresoflocation.2.Beabletocomputethemean,median,mode,quartiles,andvariouspercentiles.3.Understandthepurposeofmeasuresofvariability.4.Beabletocomputetherange,interquartilerange,variance,standarddeviation,andcoefficientofvariation.5.Understandhowzscoresarecomputedandhowtheyareusedasameasureofrelativelocationofadatavalue.4-39 6.KnowhowChebyshev’stheoremandtheempiricalrulecanbeusedtodeterminethepercentageofthedatawithinaspecifiednumberofstandarddeviationsfromthemean.7.Learnhowtoconstructa5-numbersummaryandaboxplot.8.Beabletocomputeandinterpretcovarianceandcorrelationasmeasuresofassociationbetweentwovariables.9.Beabletocomputeaweightedmean.Solutions:x75i1.x15n510,12,16,17,20Median=16(middlevalue)x96i2.x16n610,12,16,17,20,211617Median=16.523.15,20,25,25,27,28,30,3220i(8)1.62ndposition=201004-40 252025i(8)222.5100265i(8)5.26thposition=28100752830i(8)6291002x657i4.Mean59727.n11Median=576thitemMode=53Itappears3timesx1106.4i5.a.x36.88n30b.Thereareanevennumberofitems.Thus,themedianistheaverageofthe15thand16thitemsafterthedatahavebeenplacedinrankorder.36.636.7Median=36.652c.Mode=36.4Thisvalueappears4timesF25Id.FirstQuartileiHGKJ3075.100Roundingup,weseethatQ1isatthe8thposition.Q1=36.2F75Ie.ThirdQuartileiHGKJ30225.100Roundingup,weseethatQ3isatthe23rdposition.Q3=37.9x1845i6.a.x92.25n20Medianisaverageof10thand11thvaluesafterarranginginascendingorder.6695Median80.52Dataaremultimodal4-41 x1334ib.x66.7n206670Median682Mode=70(4brokerscharge$70)c.Comparingallthreemeasuresofcentrallocation(mean,medianandmode),weconcludethatitcostsmore,onaverage,totrade500sharesat$50pershare.d.Yes,trading500sharesat$50pershareisatransactionvalueof$25,000whereastrading1000sharesat$5pershareisatransactionvalueof$5000.x1380i7.a.x46n30b.Yes,themeanhereis46minutes.Thenewspaperreportedonaverageof45minutes.45529.c.Median4895.2d.Q1=7(valueof8thiteminrankedorder)Q3=70.4(valueof23rditeminrankedlist)40e.Findpositioni3012;40thpercentileisaverageofvaluesin12thand13thpositions.10040thpercentile=28.8+29.1=28.9528.a.x=775ix775ix3875.n20Themodalageis29;itappears3times.b.Medianisaverageof10thand11thitems.3740Median385.2Datasuggestat-homeworkersareslightlyyounger.c.ForQ1,25i2051004-42 Sinceiisinteger,2930Q29.512ForQ3,75i2015100Sinceiisinteger,4649Q47.53232d.i206.4100Sinceiisnotaninteger,werounduptothe7thposition.32ndpercentile=31x270,377i9.a.x10,815.08Median(Position13)=8296n25b.Medianwouldbebetterbecauseoflargedatavalues.c.i=(25/100)25=6.25Q1(Position7)=5984i=(75/100)25=18.75Q3(Position19)=14,330d.i=(85/100)25=21.2585thpercentile(position22)=15,593.Approximately85%ofthewebsiteshavelessthan15,593uniquevisitors.10.a.xi=435x435ix4833.n9Datainascendingorder:2842454849505558604-43 Median=49Donotreportamode;eachdatavalueoccursonce.Theindexcouldbeconsideredgoodsinceboththemeanandmedianarelessthan50.25b.i92.25100Q1(3rdposition)=4575i96.75100Q3(7thposition)=5552611.x26.3201516181920212222242426262727303133333458Median=25Donotreportamodesincefivevaluesappeartwice.ForQ1,25i2051002021Q20.512ForQ3,75i20151003031Q30.53212.Usingthemeanwegetx=15.58,x=18.92citycountryForthesamplesweseethatthemeanmileageisbetterinthecountrythaninthecity.City13.214.415.215.315.315.315.91616.116.216.216.716.84-44 MedianMode:15.3Country17.217.418.318.518.618.618.719.019.219.419.420.621.1MedianMode:18.6,19.4Themedianandmodalmileagesarealsobetterinthecountrythaninthecity.13.a.Mean=261/15=17.4141515151616171818181819202121MedianModeis18(occurs4times)Interpretation:theaveragenumberofcredithourstakenwas17.4.Atleast50%ofthestudentstook18ormorehours;atleast50%ofthestudentstook18orfewerhours.Themostfrequentlyoccurringnumberofcredithourstakenwas18.b.ForQ1,25i153.75100Q1(4thposition)=15ForQ3,75i1511.25100Q3(12thposition)=19c.Forthe70thpercentile,70i1510.5100Roundingupweseethe70thpercentileisinposition11.70thpercentile=184-45 x12,780i14.a.x$639n20x1976ib.x98.8picturesn20x2204ic.x110.2minutesn20d.Thisisnotaneasychoicebecauseitisamulticriteriaproblem.Ifpricewastheonlycriterion,thelowestpricecamera(FujifilmDX-10)wouldbepreferred.Ifmaximumpicturecapacitywastheonlycriterion,themaximumpicturecapacitycamera(KodakDC280Zoom)wouldbepreferred.But,ifbatterylifewastheonlycriterion,themaximumbatterylifecamera(FujifilmDX10)wouldbepreferred.Therearemanyapproachesusedtoselectthebestchoiceinamulticriteriasituation.Theseapproachesarediscussedinmorespecializedbooksondecisionanalysis.15.Range20-10=1010,12,16,17,2025i(5)1.25100Q1(2ndposition)=1275i(5)3.75100Q3(4thposition)=17IQR=Q3-Q1=17-12=5x75i16.x15n52()xx642is16n14s16417.15,20,25,25,27,28,30,34Range=34-15=19252025i(8)2Q22.511002752830i(8)6Q29110024-46 IQR=Q3-Q1=29-22.5=6.5x204ix255.n82()xx2422is3457.n17s3457..58818.a.Range=190-168=222b.(xx)376i2s=376=75.25c.s752..8678.67d.CoefficientofVariation1004.8717819.Range=92-67=25IQR=Q3-Q1=80-77=3x=78.46672xxi411.733322xxi411.7333s29.4095n114s29.40955.423120.a.Range=60-28=32IQR=Q3-Q1=55-45=10435b.x4833.92(xx)742i22()xxi742s92.75n18s92.759.634-47 c.Theaverageairqualityisaboutthesame.But,thevariabilityisgreaterinAnaheim.200021.x4005xxxx()xx2iii4104001010042040020400390400-1010040040000380400-204002000100022()xxi1000s250n14s25015.8122.DawsonSupply:Range=11-9=24.1s0.679J.C.Clark:Range=15-7=860.1s2.58923.a.WinterRange=21-12=9IQR=Q3-Q1=20-16=4SummerRange=38-18=20IQR=Q3-Q1=29-18=11b.VarianceStandardDeviationWinter8.23332.8694Summer44.48896.6700c.Winters2.8694CoefficientofVariation=10010016.21x17.7Summers6.6700CoefficientofVariation=10010026.05x25.64-48 d.Morevariabilityinthesummermonths.24.a.500Sharesat$50MinValue=34MaxValue=195Range=195-34=1614550140140QQ47.51401322Interquartilerange=140-47.5=92.51000Sharesat$5MinValue=34MaxValue=90Range=90-34=566060.579.580QQ60.2579.751322Interquartilerange=79.75-60.25=19.5b.500Sharesat$5022()xxi51,402.25s2705.3816n119s2705.381652.011000Sharesat$522()xxi5526.2s290.8526n119s290.852617.05c.500Sharesat$50s52.01CoefficientofVariation=(100)(100)56.38x92.251000Sharesat$5s17.05CoefficientofVariation=(100)(100)25.56x66.70d.Thevariabilityisgreaterforthetradeof500sharesat$50pershare.Thisistruewhetherweusethestandarddeviationorthecoefficientofvariationasameasure.225.s=0.0021Productionshouldnotbeshutdownsincethevarianceislessthan.005.4-49 26.Quartermilerss=0.0564CoefficientofVariation=(s/x)100=(0.0564/0.966)100=5.8Milerss=0.1295CoefficientofVariation=(s/x)100=(0.1295/4.534)100=2.9Yes;thecoefficientofvariationshowsthatasapercentageofthemeanthequartermilers’timesshowmorevariability.4030127.a.z210.75Atleast75%25245301b.z310.89Atleast89%25338301c.z1.610.61Atleast61%251.642301d.z2.410.83Atleast83%252.448301e.z3.610.92Atleast92%253.628.a.Approximately95%b.Almostallc.Approximately68%x75i29.x15n52()xx642is4n14101510z1.254201520z1.254121512z0.7544-50 171517z.504161516z.25452050030.z.20100650500z1.50100500500z0.00100450500z0.50100280500z2.2010031.a.Thisisfrom2standarddeviationsbelowthemeanto2standarddeviationsabovethemean.Withz=2,Chebyshev’stheoremgives:111311122z244Therefore,atleast75%ofadultssleepbetween4.5and9.3hoursperday.b.Thisisfrom2.5standarddeviationsbelowthemeanto2.5standarddeviationsabovethemean.Withz=2.5,Chebyshev’stheoremgives:111111.8422z2.56.25Therefore,atleast84%ofadultssleepbetween3.9and9.9hoursperday.c.Withz=2,theempiricalrulesuggeststhat95%ofadultssleepbetween4.5and9.3hoursperday.TheprobabilityobtainedusingtheempiricalruleisgreaterthantheprobabilityobtainedusingChebyshev’stheorem.32.a.2hoursis1standarddeviationbelowthemean.Thus,theempiricalrulesuggeststhat68%ofthekidswatchtelevisionbetween2and4hoursperday.Sinceabell-shapeddistributionissymmetric,approximately,34%ofthekidswatchtelevisionbetween2and3hoursperday.b.1houris2standarddeviationsbelowthemean.Thus,theempiricalrulesuggeststhat95%ofthekidswatchtelevisionbetween1and5hoursperday.Sinceabell-shapeddistributionissymmetric,approximately,47.5%ofthekidswatchtelevisionbetween1and3hoursperday.Inpart(a)weconcludedthatapproximately34%ofthekidswatchtelevisionbetween2and3hoursperday;thus,4-51 approximately34%ofthekidswatchtelevisionbetween3and4hoursperday.Hence,approximately47.5%+34%=81.5%ofkidswatchtelevisionbetween1and4hoursperday.c.Since34%ofthekidswatchtelevisionbetween3and4hoursperday,50%-34%=16%ofthekidswatchtelevisionmorethan4hoursperday.33.a.Approximately68%ofscoresarewithin1standarddeviationfromthemean.b.Approximately95%ofscoresarewithin2standarddeviationsfromthemean.c.Approximately(100%-95%)/2=2.5%ofscoresareover130.d.Yes,almostallIQscoresarelessthan145.71.0090.0634.a.z0.952016890.06b.z3.9020c.Thez-scoreinpartaindicatesthatthevalueis0.95standarddeviationsbelowthemean.Thez-scoreinpartbindicatesthatthevalueis3.90standarddeviationsabovethemean.Thelaborcostinpartbisanoutlierandshouldbereviewedforaccuracy.35.a.xisapproximately63or$63,000,andsis4or$4000b.Thisisfrom2standarddeviationsbelowthemeanto2standarddeviationsabovethemean.Withz=2,Chebyshev’stheoremgives:111311122z244Therefore,atleast75%ofbenefitsmanagershaveanannualsalarybetween$55,000and$71,000.c.Thehistogramofthesalarydataisshownbelow:4-52 987654Frequency321056-5858-6060-6262-6464-6666-6868-7070-7272-74SalaryAlthoughthedistributionisnotperfectlybellshaped,itdoesappearreasonabletoassumethatthedistributionofannualsalarycanbeapproximatedbyabell-shapeddistribution.d.Withz=2,theempiricalrulesuggeststhat95%ofbenefitsmanagershaveanannualsalarybetween$55,000and$71,000.TheprobabilityismuchhigherthanobtainedusingChebyshev’stheorem,butrequirestheassumptionthatthedistributionofannualsalaryisbellshaped.e.Therearenooutliersbecausealltheobservationsarewithin3standarddeviationsofthemean.36.a.xis100andsis13.88orapproximately14b.Ifthedistributionisbellshapedwithameanof100points,thepercentageofNBAgamesinwhichthewinningteamscoresmorethan100pointsis50%.Ascoreof114pointsisz=1standarddeviationabovethemean.Thus,theempiricalrulesuggeststhat68%ofthewinningteamswillscorebetween86and114points.Inotherwords,32%ofthewinningteamswillscorelessthan86pointsormorethan114points.Becauseabell-shapeddistributionissymmetric,approximately16%ofthewinningteamswillscoremorethan114points.c.Forthewinningmargin,xis11.1andsis10.77.Toseeifthereareanyoutliers,wewillfirstcomputethez-scoreforthewinningmarginthatisfarthestfromthesamplemeanof11.1,awinningmarginof32points.xx3211.1z1.94s10.77Thus,awinningmarginof32pointsisnotanoutlier(z=1.94<3).Becauseawinningmarginof32pointsisfarthestfromthemean,noneoftheotherdatavaluescanhaveaz-scorethatislessthan3orgreaterthan3andhenceweconcludethattherearenooutliersx7986.i37.a.x399.n204-53 4.174.20Median=4.185(averageof10thand11thvalues)2b.Q1=4.00(averageof5thand6thvalues)Q3=4.50(averageof15thand16thvalues)2()xx125080.ic.s08114.n119412399..d.AllisonOne:z016.08114.232399..OmniAudioSA12.3:z206.08114.e.ThelowestratingisfortheBose501Series.It’sz-scoreis:214399..z228.08114.Thisisnotanoutliersotherearenooutliers.38.15,20,25,25,27,28,30,34Smallest=15252025i()82Q225.110022527Median262752830i()88Q2931002Largest=3439.152025303540.5,6,8,10,10,12,15,16,18Smallest=525i(9)2.25Q1=8(3rdposition)1004-54 Median=1075i(9)6.75Q3=15(7thposition)100Largest=18510152041.IQR=50-42=8LowerLimit:Q1-1.5IQR=42-12=30UpperLimit:Q3+1.5IQR=50+12=6268isanoutlier42.a.Fivenumbersummary:59.614.519.252.7b.IQR=Q3-Q1=19.2-9.6=9.6LowerLimit:Q1-1.5(IQR)=9.6-1.5(9.6)=-4.8UpperLimit:Q3+1.5(IQR)=19.2+1.5(9.6)=33.6c.Thedatavalue41.6isanoutlier(largerthantheupperlimit)andsoisthedatavalue52.7.Thefinancialanalystshouldfirstverifythatthesevaluesarecorrect.Perhapsatypingerrorhascaused25.7tobetypedas52.7(or14.6tobetypedas41.6).Iftheoutliersarecorrect,theanalystmightconsiderthesecompanieswithanunusuallylargereturnonequityasgoodinvestmentcandidates.d.**-1052035506543.a.Median(11thposition)401925i(21)5.25100Q1(6thposition)=187275i(21)15.75100Q3(16thposition)=8305608,1872,4019,8305,141384-55 b.Limits:IQR=Q3-Q1=8305-1872=6433LowerLimit:Q1-1.5(IQR)=-7777UpperLimit:Q3+1.5(IQR)=17955c.Therearenooutliers,alldataarewithinthelimits.d.Yes,ifthefirsttwodigitsinJohnsonandJohnson"ssalesweretransposedto41,138,saleswouldhaveshownupasanoutlier.Areviewofthedatawouldhaveenabledthecorrectionofthedata.e.03,0006,0009,00012,00015,00044.a.Mean=105.7933Median=52.7b.Q1=15.7Q3=78.3c.IQR=Q3-Q1=78.3-15.7=62.6Lowerlimitforboxplot=Q1-1.5(IQR)=15.7-1.5(62.6)=-78.2Upperlimitforboxplot=Q3+1.5(IQR)=78.3+1.5(62.6)=172.2Note:Becausethenumberofsharescoveredbyoptionsgrantscannotbenegative,thelowerlimitfortheboxplotissetat0.This,outliersarevalueinthedatasetgreaterthan172.2.Outliers:SiliconGraphics(188.8)andToysRUs(247.6)d.Meanpercentage=26.73.Thecurrentpercentageismuchgreater.45.a.FiveNumberSummary(Midsize)5171.581.596.5128FiveNumberSummary(Small)73101108.5121140b.BoxPlotsMidsize4-56 5060708090100110120130SmallSize5060708090100110120130140150c.Themidsizecarsappeartobesaferthanthesmallcars.46.a.x=37.48Median=23.67b.Q1=7.91Q3=51.92c.IQR=51.92-7.91=44.01LowerLimit:Q1-1.5(IQR)=7.91-1.5(44.01)=-58.11UpperLimit:Q3+1.5(IQR)=51.92+1.5(44.01)=117.94Russia,withapercentchangeof125.89,isanoutlier.Turkey,withapercentchangeof254.45isanotheroutlier.d.Withapercentchangeof22.64,theUnitedStatesisjustbelowthe50thpercentile-themedian.47.a.70605040y302010005101520xb.Negativerelationship4-57 40230c/d.xx408yy23046ii5522()xxyy()240()xx118()yy520iiii()xxyy()240iis60xyn1512()xx118is5.4314xn1512()yy520is11.4018yn151sxy60r0.969xyss(5.4314)(11.4018)xyThereisastrongnegativelinearrelationship.48.a.1816141210y86420051015202530xb.Positiverelationship8050c/d.xx8016yy5010ii5522()xxyy()106()xx272()yy86iiii()xxyy()106iis26.5xyn1514-58 2()xx272is8.2462xn1512()yy86is4.6368yn151sxy26.5r0.693xyss(8.2462)(4.6368)xyApositivelinearrelationship49.a.750700650600550y=SAT5004504002.62.833.23.43.63.8x=GPAb.Positiverelationship198.3540c/d.xx198.33.yy3540590ii6622()xxyy()143().(),xx074yy36400iiii()xxyy()143iis28.6xyn1612()xx0.74is0.3847xn1612()yy36,400is85.3229yn161sxy28.6r0.8713xyss(0.3847)(85.3229)xyApositivelinearrelationship4-59 50.Letx=drivingspeedandy=mileage420270xx42042yy27027ii101022(xxyy)()475(xx)1660(yy)164iiii()xxyy()475iis52.7778xyn11012()xx1660is13.5810xn11012()yy164is4.2687yn1101sxy52.7778r.91xyss(13.5810)(4.2687)xyAstrongnegativelinearrelationship51.a.Thesamplecorrelationcoefficientis.78.b.Thereisapositivelinearrelationshipbetweentheperformancescoreandtheoverallrating.52.a.Thesamplecorrelationcoefficientis.92.b.Thereisastrongpositivelinearrelationshipbetweenthetwovariables.53.Thesamplecorrelationcoefficientis.88.Thisindicatesastrongpositivelinearrelationshipbetweenthedailyhighandlowtemperatures.wx6323222585(.)()(.)()702.ii54.a.x369.w632819i322255...127b.3175.4455.fiMifiMi45207107091513552010025325fM325iix13n254-60 fiMiMx()Mx2f()Mx2iiii45-864256710-3963915+2436520+74924560022fMxii()600s25n124s25556.a.GradexiWeightWi4(A)93(B)152(C)331(D)30(F)060CreditHourswx9415333231()()()()150iix250.w91533360ib.Yes;satisfiesthe2.5gradepointaveragerequirement57.Weusetheweightedmeanformulawiththeweightsbeingtheamountsinvested.wx=37,830(0.00)+27,667(2.98)+31,037(2.77)+27,336(2.65)+37,553(1.58)ii+17,812(0.57)+32,660(2.00)+17,775(0.00)=375,667.1w=37,830+27,667+···+17,775i=229,670wx3756671,.iix164.w229670,i58.MififiMiMix()Mx2f()Mx2iii742148-8.74264776.4338775,656.106919271,344-3.74264714.0074072,689.4221280123,3601.2573531.580937442.6622105171,7856.25735339.1544674,111.2190232250611.257353126.7280002,914.743962716216.257353264.3015301,585.80926807,30517,399.9630Estimateoftotalgallonssold:(10.74)(120)=1288.87305x10.746804-61 217,399.9630s25.63679s5.0659.a.ClassfiMifiMi015001101102402803853255435041400Totals5001745fM1745iix349.n500b.Mx()Mx22iifMxii()-3.4912.18182.70-2.496.2062.00-1.492.2288.80-0.490.2420.41+0.510.2691.04Total444.952()Mxf44495.2iis08917..s0891709443.n1499x3463i60.a.x13852.n25Median=129(13thvalue)Mode=0(2times)b.ItappearsthatthisgroupofyoungadultseatsoutmuchmorethantheaverageAmerican.Themeanandmedianaremuchhigherthantheaverageof$65.88reportedinthenewspaper.c.Q1=95(7thvalue)Q3=169(19thvalue)d.Min=0Max=467Range=467-0=467IQR=Q3-Q1=169-95=742e.s=9271.01s=96.294-62 f.Thez-scoreforthelargestvalueis:46713852.z341.9629.Itistheonlyoutlierandshouldbecheckedforaccuracy.61.a.xi=760x760ix38n20Medianisaverageof10thand11thitems.3636Median362Themodalcashretaineris40;itappears4times.b.ForQ1,c.25i205100Sinceiisinteger,2830Q2912ForQ3,75i2015100Sinceiisinteger,4050Q4532cRange=64–15=49Interquartilerange=45–29=1622xxi3318d.s174.6316n12012ss174.631613.2148s13.2148e.Coefficientofvariation=10010034.8x384-63 x260i62.a.x1857.n14Median=16.5(Averageof7thand8thvalues)2b.s=53.49s=7.31c.Quantexhasthebestrecord:11Days271857.d.z115.731.Packard-Bellis1.15standarddeviationsslowerthanthemean.121857.e.z090.731.IBMis0.9standarddeviationsfasterthanthemean.f.CheckToshiba:371857.z252.731.Onthebasisofz-scores,Toshibaisnotanoutlier,butitis2.52standarddeviationsslowerthanthemean.63.x=1890.2/30=63Median(15thand16thpositions)is(63+63.5)/2=63.25Mode:60.5and63.5bothoccurtwiceb.i=(25/100)30=7.5(8thposition)Q1=55.9i=(75/100)30=22.5(23rdposition)Q3=69.064.Samplemean=7195.5Median=7019(averageofpositions5and6)Samplevariance=7,165,941Samplestandarddeviation=2676.9365.a.Thesamplemeanis83.135andthesamplestandarddeviationis16.173.b.Withz=2,Chebyshev’stheoremgives:4-64 111311122z244Therefore,atleast75%ofhouseholdincomesarewithin2standarddeviationsofthemean.Usingthesamplemeanandsamplestandarddeviationcomputedinpart(a),therangewithin75%ofhouseholdincomesmustfallis83.1352(16.173)=83.13532.346;thus,75%ofhouseholdincomesmustfallbetween50.789and115.481,or$50,789to$115,481.c.Withz=2,theempiricalrulesuggeststhat95%ofhouseholdincomesmustfallbetween$50,789to$115,481.Forthesamerange,theprobabilityobtainedusingtheempiricalruleisgreaterthantheprobabilityobtainedusingChebyshev’stheorem.d.Thez-scoreforDanbury,CTis3.04;thus,theDanbury,CTobservationisanoutlier.32066.a.PublicTransportation:x3210320Automobile:x3210b.PublicTransportation:s=4.64Automobile:s=1.83c.Prefertheautomobile.Themeantimesarethesame,buttheautohaslessvariability.d.Datainascendingorder:Public:25282929323233343741Auto:29303131323233333435FivenumberSummariesPublic:2529323441Auto:2931323335BoxPlots:Public:2428323640Auto:4-65 2428323640Theboxplotsdoshowlowervariabilitywithautomobiletransportationandsupporttheconclusioninpartc.67.Datainascendingorder:42445356586162627576777879828484858889899395969798a.FiveNumberSummary4262798998b.BoxPlot40506070809010068.Datainascendingorder:40045151157659665271174480982085290794197197510231112117412511278i=(25/100)20=5i=(75/100)20=15i=(50/100)20=10596652Q624129751023Q99932820852Median=8362a.FiveNumberSummary4-66 4006248369991278b.4005006007008009001000110012001300c.Therearenovaluesoutsidethelimits.Thusnooutliersareidentified.Lowerlimit=624-1.5(999-624)=61.5Upperlimit=999+1.5(999-624)=1561.569.a.Thesamplecovarianceis477.5365.Becausethesamplecovarianceispositive,thereisapositivelinearrelationshipbetweenincomeandhomeprice.b.Thesamplecorrelationcoefficientis.933;thisindicatesastronglinearrelationshipbetweenincomeandhomeprice.70.a.Thescatterdiagramindicatesapositiverelationshipb.xy79811,688xy1,058,019iiii22xy71,30616,058,736iixyiixyii/n1,058,019(798)(11,688)/9r.9856xy222222xxnyynii//ii71,306(798)/916,058,736(11,688)/9Strongpositiverelationship71.Letxi=commissionon500sharestradeforbrokeriyi=commissionon1000sharestradeforbrokeri18291326xx182991.45yy132666.3ii202022(xxyy)()11,853.3(xx)48,370.95(yy)8506.2iiii()xxyy()11,853.3iis623.8579xyn119Thecovarianceshowsthereisapositiverelationship.2()xx48,370.95is50.4563xn1194-67 2()yy8506.2is21.1588yn119sxy623.8579r0.5844xyss(50.4563)(21.1588)xyThecorrelationcoefficientshowsthatwhiletherelationshipispositive,itisnotrealstrong.NotethatMaxUlechargesmorethanSchwabforthe500sharetrade($195vs.$155)butlessforthe1000sharetrade($70vs.$90).72.a.Thescatterdiagramisshownbelow:3.532.52Earnings1.510.50051015202530BookValueb.Thesamplecorrelationcoefficientis.75;thisindicatesalinearrelationshipbetweenbookvalueandearnings.wx20(20)30(12)10(7)15(5)10(6)965ii73.x11.4daysw203010151085i74.a.(800+750+900)/3=817b.MonthJanuaryFebruaryMarchWeight123wx1800()()()275039005000iix833w1236i75.fiMifiMiMxi()Mx2fMx()2iii4-68 45.522.0-6.846.24184.9659.547.5-2.87.8439.20713.594.51.21.4410.08217.535.05.227.0454.08121.521.59.284.6484.64125.525.513.2174.24174.2420246.0547.20246x12.3202547.20s28.819s=5.3776.fiMifiMiMxi()Mx2fMx()2iii229.559.0-22484968639.5237.0-12144864449.5198.0-2416459.5238.0864256269.5139.018324648279.5159.0287841568201,030.043201030x51.5204320s227.3719s=15.0877.fiMifiMiMxi()Mx2fMx()2iii1047470-13.68187.14241871.4240522080-8.6875.34243013.70150578550-3.6813..54242031.361756210850+1.321.7424304.9275675025+6.3239.94242995.6815721080+11.32128.14241922.141077770+16.32266.34242663.4247528,82514,802.6428,825a.x60.68475214,802.64b.s31.23474s31.235.594-69 Chapter4IntroductiontoProbabilityLearningObjectives1.Obtainanappreciationoftheroleprobabilityinformationplaysinthedecisionmakingprocess.2.Understandprobabilityasanumericalmeasureofthelikelihoodofoccurrence.3.Knowthethreemethodscommonlyusedforassigningprobabilitiesandunderstandwhentheyshouldbeused.4.Knowhowtousethelawsthatareavailableforcomputingtheprobabilitiesofevents.5.Understandhownewinformationcanbeusedtoreviseinitial(prior)probabilityestimatesusingBayes’theorem.4-70 Solutions:1.NumberofexperimentalOutcomes=(3)(2)(4)=24F6I6!6543212.20HG3KJ33!!(321321)()ABCACEBCDBEFABDACFBCECDEABEADEBCFCDFABFADFBDECEFACDAEFBDFDEF66!3.P()()()6541203()63!BDFBFDDBFDFBFBDFDB4.a.1stToss2ndToss3rdTossH(H,H,H)TH(H,H,T)TH(H,T,H)HT(H,T,T)TH(T,H,H)TH(T,H,T)TH(T,T,H)T(T,T,T)b.Let:HbeheadandTbetail(H,H,H)(T,H,H)(H,H,T)(T,H,T)(H,T,H)(T,T,H)(H,T,T)(T,T,T)c.Theoutcomesareequallylikely,sotheprobabilityofeachoutcomesis1/8.5.P(Ei)=1/5fori=1,2,3,4,5P(Ei)0fori=1,2,3,4,5P(E1)+P(E2)+P(E3)+P(E4)+P(E5)=1/5+1/5+1/5+1/5+1/5=1Theclassicalmethodwasused.13-71 6.P(E1)=.40,P(E2)=.26,P(E3)=.34Therelativefrequencymethodwasused.7.No.Requirement(4.3)isnotsatisfied;theprobabilitiesdonotsumto1.P(E1)+P(E2)+P(E3)+P(E4)=.10+.15+.40+.20=.858.a.Therearefouroutcomespossibleforthis2-stepexperiment;planningcommissionpositive-councilapproves;planningcommissionpositive-councildisapproves;planningcommissionnegative-councilapproves;planningcommissionnegative-councildisapproves.b.Letp=positive,n=negative,a=approves,andd=disapprovesPlanningCommissionCouncil(p,a)adp(p,d).n(n,a)ad(n,d)F50I50!504948479.230300,HG4KJ446!!432110.a.Usetherelativefrequencyapproach:P(California)=1,434/2,374=.60b.Numbernotfrom4states=2,374-1,434-390-217-112=221P(Notfrom4States)=221/2,374=.09c.P(NotinEarlyStages)=1-.22=.78d.EstimateofnumberofMassachusettscompaniesinearlystageofdevelopment-(.22)3908613-72 e.Ifweassumethesizeoftheawardsdidnotdifferbystates,wecanmultiplytheprobabilityanawardwenttoColoradobythetotalventurefundsdisbursedtogetanestimate.EstimateofColoradofunds=(112/2374)($32.4)=$1.53billionAuthors"Note:TheactualamountgoingtoColoradowas$1.74billion.11.a.No,theprobabilitiesdonotsumtoone.Theysumto.85.b.Ownermustrevisetheprobabilitiessotheysumto1.00.12.a.Usethecountingruleforcombinations:F49I49!()4948474645()()()()1906884,,HG5KJ544!!()()()()()54321b.Verysmall:1/1,906,884=0.0000005c.Multiplytheanswertopart(a)by42togetthenumberofchoicesforthesixnumbers.No.ofChoices=(1,906,884)(42)=80,089,128ProbabilityofWinning=1/80,089,128=0.000000012513.Initiallyaprobabilityof.20wouldbeassignedifselectionisequallylikely.Datadoesnotappeartoconfirmthebeliefofequalconsumerpreference.Forexampleusingtherelativefrequencymethodwewouldassignaprobabilityof5/100=.05tothedesign1outcome,.15todesign2,.30todesign3,.40todesign4,and.10todesign5.14.a.P(E2)=1/4b.P(any2outcomes)=1/4+1/4=1/2c.P(any3outcomes)=1/4+1/4+1/4=3/415.a.S={aceofclubs,aceofdiamonds,aceofhearts,aceofspades}b.S={2ofclubs,3ofclubs,...,10ofclubs,Jofclubs,Qofclubs,Kofclubs,Aofclubs}c.Thereare12;jack,queen,orkingineachofthefoursuits.d.Fora:4/52=1/13=.08Forb:13/52=1/4=.25Forc:12/52=.2313-73 16.a.(6)(6)=36samplepointsb.Die212345612345672345678TotalforBoth.3456789Die1456789105678910116789101112c.6/36=1/6d.10/36=5/18e.No.P(odd)=18/36=P(even)=18/36or1/2forboth.f.Classical.Aprobabilityof1/36isassignedtoeachexperimentaloutcome.17.a.(4,6),(4,7),(4,8)b..05+.10+.15=.30c.(2,8),(3,8),(4,8)d..05+.05+.15=.25e..1518.a.0;probabilityis.05b.4,5;probabilityis.10+.10=.20c.0,1,2;probabilityis.05+.15+.35=.5519.a.Yes,theprobabilitiesareallgreaterthanorequaltozeroandtheysumtoone.b.P(A)=P(0)+P(1)+P(2)=.08+.18+.32=.5813-74 c.P(B)=P(4)=.1220.a.P(N)=56/500=.112b.P(T)=43/500=.086c.Totalin6states=56+53+43+37+28+28=245P(B)=245/500=.49AlmosthalftheFortune500companiesareheadquarteredinthesestates.21.a.P(A)=P(1)+P(2)+P(3)+P(4)+P(5)2012631=5050505050=.40+.24+.12+.06+.02=.84b.P(B)=P(3)+P(4)+P(5)=.12+.06+.02=.20c.P(2)=12/50=.2422.a.P(A)=.40,P(B)=.40,P(C)=.60b.P(AB)=P(E1,E2,E3,E4)=.80.YesP(AB)=P(A)+P(B).ccccc.A={E3,E4,E5}C={E1,E4}P(A)=.60P(C)=.40ccd.AB={E1,E2,E5}P(AB)=.60e.P(BC)=P(E2,E3,E4,E5)=.8023.a.P(A)=P(E1)+P(E4)+P(E6)=.05+.25+.10=.40P(B)=P(E2)+P(E4)+P(E7)=.20+.25+.05=.50P(C)=P(E2)+P(E3)+P(E5)+P(E7)=.20+.20+.15+.05=.60b.AB={E1,E2,E4,E6,E7}P(AB)=P(E1)+P(E2)+P(E4)+P(E6)+P(E7)=.05+.20+.25+.10+.05=.65c.AB={E4}P(AB)=P(E4)=.2513-75 d.Yes,theyaremutuallyexclusive.cce.B={E1,E3,E5,E6};P(B)=P(E1)+P(E3)+P(E5)+P(E6)=.05+.20+.15+.10=.5024.P(CrashNotLikely)=1-.14-.43=.4325.LetY=highone-yearreturnM=highfive-yearreturna.P(Y)=15/30=.50P(M)=12/30=.40P(YM)=6/30=.20b.P(YM)=P(Y)+P(M)-P(YM)=.50+.40-.20=.70c.1-P(YM)=1-.70=.3026.LetY=highone-yearreturnM=highfive-yearreturna.P(Y)=9/30=.30P(M)=7/30=.23b.P(YM)=5/30=.17c.P(YM)=.30+.23-.17=.36P(Neither)=1-.36=.6427.Let:D=consumesorservesdomesticwineI=consumesorservesimportedwineWearegivenP(D)=0.57,P(I)=0.33,P(DI)=0.63P(DI)=P(D)+P(I)-P(DI)=0.57+0.33-0.63=0.2728.Let:B=rentedacarforbusinessreasonsP=rentedacarforpersonalreasonsa.P(BP)=P(B)+P(P)-P(BP)=.54+.458-.30=.698b.P(Neither)=1-.698=.30213-76 725790,29.a.P(H)0299.2425000,,537390,P(C)0222.2425000,,159877,P(S)0066.2425000,,b.Apersoncanhaveonlyoneprimarycauseofdeathlistedonadeathcertificate.So,theyaremutuallyexclusive.c.P(HC)=0.299+0.222=0.521d.P(CS)=0.222+0.066=0.288e.1-0.299-0.222-0.066=0.413P(AB).4030.a.P(AB).6667P(B).60P(AB).40b.P(BA).80P(A).50c.NobecauseP(A|B)P(A)31.a.P(AB)=0P(AB)0b.P(AB)0P(B).4c.No.P(A|B)P(A);theevents,althoughmutuallyexclusive,arenotindependent.d.Mutuallyexclusiveeventsaredependent.32.a.SingleMarriedTotalUnder30.55.10.6530orover.20.15.35Total.75.251.00b.65%ofthecustomersareunder30.c.Themajorityofcustomersaresingle:P(single)=.75.13-77 d..55e.Let:A=eventunder30B=eventsingleP(AB).55P(BA).8462P(A).65f.P(AB)=.55P(A)P(B)=(.65)(.75)=.49SinceP(AB)P(A)P(B),theycannotbeindependentevents;or,sinceP(A|B)P(B),theycannotbeindependent.33.a.ReasonforApplyingQualityCost/ConvenienceOtherTotalFullTime.218.204.039.461PartTime.208.307.024.539.426.511.0631.00b.Itismostlikelyastudentwillcitecostorconvenienceasthefirstreason-probability=.511.Schoolqualityisthefirstreasoncitedbythesecondlargestnumberofstudents-probability=.426.c.P(Quality|fulltime)=.218/.461=.473d.P(Quality|parttime)=.208/.539=.386e.Forindependence,wemusthaveP(A)P(B)=P(AB).Fromthetable,P(AB)=.218,P(A)=.461,P(B)=.426P(A)P(B)=(.461)(.426)=.196SinceP(A)P(B)P(AB),theeventsarenotindependent.34.a.P(O)=0.38+0.06=0.44b.P(Rh-)=0.06+0.02+0.01+0.06=0.15c.P(bothRh-)=P(Rh-)P(Rh-)=(0.15)(0.15)=0.022513-78 d.P(bothAB)=P(AB)P(AB)=(0.05)(0.05)=0.0025P(Rh-O).06e.P(Rh-O).136P(O).44f.P(Rh+)=1-P(Rh-)=1-0.15=0.85P(BRh+).09P(BRh+).106P(Rh+).8535.a.P(UpforJanuary)=31/48=0.646b.P(UpforYear)=36/48=0.75c.P(UpforYearUpforJanuary)=29/48=0.604P(UpforYear|UpforJanuary)=0.604/0.646=0.935d.TheyarenotindependentsinceP(UpforYear)P(UpforYear|UpforJanuary)0.750.93536.a.SatisfactionScoreOccupationUnder5050-5960-6970-7980-89TotalCabinetmaker.000.050.100.075.025.250Lawyer.150.050.025.025.000.250PhysicalTherapist.000.125.050.025.050.250SystemsAnalyst.050.025.100.075.000.250Total.200.250.275.200.0751.000b.P(80s)=.075(amarginalprobability)c.P(80s|PT)=.050/.250=.20(aconditionalprobability)d.P(L)=.250(amarginalprobability)e.P(LUnder50)=.150(ajointprobability)f.P(Under50|L)=.150/.250=.60(aconditionalprobability)g.P(70orhigher)=.275(Sumofmarginalprobabilities)37.a.P(AB)=P(A)P(B)=(.55)(.35)=.19b.P(AB)=P(A)+P(B)-P(AB)=.55+.35-.19=.71c.P(shutdown)=1-P(AB)=1-.71=.2913-79 5238.a.P(Telephone)0.2737190b.Thisisanintersectionoftwoevents.Itseemsreasonabletoassumethenexttwomessageswillbeindependent;weusethemultiplicationruleforindependentevents.3015P(E-mailFax)=P(E-mail)P(Fax)=0.0125190190c.Thisisaunionoftwomutuallyexclusiveevents.P(TelephoneInterofficeMail)=P(Telephone)+P(InterofficeMail)521870=0.736819019019039.a.Yes,sinceP(A1A2)=0b.P(A1B)=P(A1)P(B|A1)=.40(.20)=.08P(A2B)=P(A2)P(B|A2)=.60(.05)=.03c.P(B)=P(A1B)+P(A2B)=.08+.03=.11.08d.P(AB).72731.11.03P(AB).27272.1140.a.P(BA1)=P(A1)P(B|A1)=(.20)(.50)=.10P(BA2)=P(A2)P(B|A2)=(.50)(.40)=.20P(BA3)=P(A3)P(B|A3)=(.30)(.30)=.09.20b.P(AB).512.10.20.09c.EventsP(Ai)P(B|Ai)P(AiB)P(Ai|B)A1.20.50.10.26A2.50.40.20.51A3.30.30.09.231.00.391.0041.S1=successful,S2=notsuccessfulandB=requestreceivedforadditionalinformation.a.P(S1)=.50b.P(B|S1)=.7513-80 (.50)(.75).375c.P(SB).651(.50)(.75)(.50)(.40).57542.M=missedpaymentD1=customerdefaultsD2=customerdoesnotdefaultP(D1)=.05P(D2)=.95P(M|D2)=.2P(M|D1)=1P(D)P(MD)(.05)(1).0511a.P(DM).211P(D)P(MD)P(D)P(MD)(.05)(1)(.95)(.2).241122b.Yes,theprobabilityofdefaultisgreaterthan.20.43.Let:S=smallcarcS=othertypeofvehicleF=accidentleadstofatalityforvehicleoccupantccWehaveP(S)=.18,soP(S)=.82.AlsoP(F|S)=.128andP(F|S)=.05.UsingthetabularformofBayesTheoremprovides:PriorConditionalJointPosteriorEventsProbabilitiesProbabilitiesProbabilitiesProbabilitiesS.18.128.023.36cS.82.050.041.641.00.0641.00Fromtheposteriorprobabilitycolumn,wehaveP(S|F)=.36.So,ifanaccidentleadstoafatality,theprobabilityasmallcarwasinvolvedis.36.44.LetA1=StoryaboutBasketballTeamA2=StoryaboutHockeyTeamW="WeWin"headlineP(A1)=.60P(W|A1)=.641P(A2)=.40P(W|A2)=.462AiP(Ai)P(W|A1)P(WAi)P(Ai|M)A1.60.641.3846.3846/.5694=.6754A2.40.462.1848.1848/.5694=.3246.56941.0000Theprobabilitythestoryisaboutthebasketballteamis.6754.45.a.EventsP(Di)P(S1|Di)P(DiS1)P(Di|S1)D1.60.15.090.2195D2.40.80.320.78051.00P(S1)=.4101.000P(D1|S1)=.219513-81 P(D2|S1)=.7805b.EventsP(Di)P(S2|Di)P(DiS2)P(Di|S2)D1.60.10.060.500D2.40.15.060.5001.00P(S2)=.1201.000P(D1|S2)=.50P(D2|S2)=.50c.EventsP(Di)P(S3|Di)P(DiS3)P(Di|S3)D1.60.15.090.8824D2.40.03.012.11761.00P(S3)=.1021.0000P(D1|S3)=.8824P(D2|S3)=.1176d.Usetheposteriorprobabilitiesfrompart(a)asthepriorprobabilitieshere.EventsP(Di)P(S2|Di)P(DiS2)P(Di|S2)D1.2195.10.0220.1582D2.7805.15.1171.84181.0000.13911.0000P(D1|S1andS2)=.1582P(D2|S1andS2)=.841846.a.P(Excellent)=.18P(PrettyGood)=.50P(PrettyGoodExcellent)=.18+.50=.68Note:Eventsaremutuallyexclusivesinceapersonmayonlychooseonerating.b.1035(.05)=51.75Weestimate52respondentsratedUScompaniespoor.c.1035(.01)=10.35Weestimate10respondentsdidnotknowordidnotanswer.47.a.(2)(2)=4b.Lets=successfulu=unsuccessful13-82 OilBondsE1susE2uE3suE4c.O={E1,E2}M={E1,E3}d.OM={E1,E2,E3}e.OM={E1}f.No;sinceOMhasasamplepoint.48.a.P(satisfied)=0.61b.The18-34yearoldgroup(64%satisfied)andthe65andovergroup(70%satisfied).c.P(notsatisfied)=0.26+0.04=0.3049.LetI=treatment-causedinjuryD=deathfrominjuryN=injurycausedbynegligenceM=malpracticeclaimfiled$=paymentmadeinclaimWearegivenP(I)=0.04,P(N|I)=0.25,P(D|I)=1/7,P(M|N)=1/7.5=0.1333,andP($|M)=0.50cca.P(N)=P(N|I)P(I)+P(N|I)P(I)=(0.25)(0.04)+(0)(0.96)=0.0113-83 ccb.P(D)=P(D|I)P(I)+P(D|I)P(I)=(1/7)(0.04)+(0)(0.96)=0.006ccc.P(M)=P(M|N)P(N)+P(M|N)P(N)=(0.1333)(0.01)+(0)(0.99)=0.001333ccP($)=P($|M)P(M)+P($|M)P(M)=(0.5)(0.001333)+(0)(0.9987)=0.0006750.a.Probabilityoftheevent=P(average)+P(aboveaverage)+P(excellent)111413=505050=.22+.28+.26=.76b.Probabilityoftheevent=P(poor)+P(belowaverage)48=.24505051.a.P(leases1)=168/932=0.18b.P(2orfewer)=401/932+242/932+65/932=708/932=0.76c.P(3ormore)=186/932+112/932=298/932=0.32d.P(nocars)=19/932=0.0252.a.13-84 YesNoTotal23andUnder.1026.0996.202224-26.1482.1878.336027-30.0917.1328.224531-35.0327.0956.128336andOver.0253.0837.1090Total.4005.59951.0000.b..2022c..2245+.1283+.1090=.4618d..400553.a.P(24to26|Yes)=.1482/.4005=.3700b.P(Yes|36andover)=.0253/.1090=.2321c..1026+.1482+.1878+.0917+.0327+.0253=.5883d.P(31ormore|No)=(.0956+.0837)/.5995=.2991e.No,becausetheconditionalprobabilitiesdonotallequalthemarginalprobabilities.Forinstance,P(24to26|Yes)=.3700P(24to26)=.336054.LetI=importantorveryimportantM=maleF=femalea.P(I)=.49(amarginalprobability)b.P(I|M)=.22/.50=.44(aconditionalprobability)c.P(I|F)=.27/.50=.54(aconditionalprobability)d.ItisnotindependentP(I)=.49P(I|M)=.44andP(I)=.49P(I|F)=.5413-85 e.Sincelevelofimportanceisdependentongender,weconcludethatmaleandfemalerespondentshavedifferentattitudestowardrisk.P(BS).1255.a.P(BS).30P(S).40WehaveP(B|S)>P(B).Yes,continuetheadsinceitincreasestheprobabilityofapurchase.b.Estimatethecompany’smarketshareat20%.ContinuingtheadvertisementshouldincreasethemarketsharesinceP(B|S)=.30.P(BS).10c.P(BS).333P(S).30Thesecondadhasabiggereffect.56.a.P(A)=200/800=.25b.P(B)=100/800=.125c.P(AB)=10/800=.0125d.P(A|B)=P(AB)/P(B)=.0125/.125=.10e.No,P(A|B)P(A)=.2557.LetA=losttimeaccidentincurrentyearB=losttimeaccidentpreviousyearGiven:P(B)=.06,P(A)=.05,P(A|B)=.15a.P(AB)=P(A|B)P(B)=.15(.06)=.009b.P(AB)=P(A)+P(B)-P(AB)=.06+.05-.009=.101or10.1%58.Let:A=returnisfraudulentB=exceedsIRSstandardfordeductionscGiven:P(A|B)=.20,P(A|B)=.02,P(B)=.08,findP(A)=.3.cNoteP(B)=1-P(B)=.92cP(A)=P(AB)+P(AB)cc=P(B)P(A|B)+P(B)P(A|B)=(.08)(.20)+(.92)(.02)=.0344Weestimate3.44%willbefraudulent.59.a.P(Oil)=.50+.20=.70b.LetS=Soiltestresults13-86 EventsP(Ai)P(S|Ai)P(AiS)P(Ai|S)HighQuality(A1).50.20.10.31MediumQuality(A2).20.80.16.50NoOil(A3).30.20.06.191.00P(S)=.321.00P(Oil)=.81whichisgood;however,probabilitiesnowfavormediumqualityratherthanhighqualityoil.60.a.A1=fieldwillproduceoilA2=fieldwillnotproduceoilW=wellproducesoilcccEventsP(Ai)P(W|Ai)P(WAi)P(Ai|W)OilinField.25.20.05.0625NoOilinField.751.00.75.93751.00.801.0000Theprobabilitythefieldwillproduceoilgivenawellcomesupdryis.0625.b.cccEventsP(Ai)P(W|Ai)P(WAi)P(Ai|W)OilinField.0625.20.0125.0132NoOilinField.93751.00.9375.98681.0000.95001.0000Theprobabilitythewellwillproduceoildropsfurtherto.0132.c.Supposeathirdwellcomesupdry.Theprobabilitiesarerevisedasfollows:cccEventsP(Ai)P(W|Ai)P(WAi)P(Ai|W)OilinField.0132.20.0026.0026IncorrectAdjustment.98681.00.9868.99741.0000.98941.0000Stopdrillingandabandonfieldifthreeconsecutivewellscomeupdry.Chapter5DiscreteProbabilityDistributionsLearningObjectives1.Understandtheconceptsofarandomvariableandaprobabilitydistribution.2.Beabletodistinguishbetweendiscreteandcontinuousrandomvariables.13-87 3.Beabletocomputeandinterprettheexpectedvalue,variance,andstandarddeviationforadiscreterandomvariable.4.Beabletocomputeandworkwithprobabilitiesinvolvingabinomialprobabilitydistribution.5.BeabletocomputeandworkwithprobabilitiesinvolvingaPoissonprobabilitydistribution.6.Knowwhenandhowtousethehypergeometricprobabilitydistribution.Solutions:1.a.Head,Head(H,H)Head,Tail(H,T)Tail,Head(T,H)Tail,Tail(T,T)b.x=numberofheadsontwocointossesc.OutcomeValuesofx(H,H)2(H,T)1(T,H)113-88 (T,T)0d.Discrete.Itmayassume3values:0,1,and2.2.a.Letx=time(inminutes)toassembletheproduct.b.Itmayassumeanypositivevalue:x>0.c.Continuous3.LetY=positionisofferedN=positionisnotoffereda.S={(Y,Y,Y),(Y,Y,N),(Y,N,Y),(Y,N,N),(N,Y,Y),(N,Y,N),(N,N,Y),(N,N,N)}b.LetN=numberofoffersmade;Nisadiscreterandomvariable.c.ExperimentalOutcome(Y,Y,Y)(Y,Y,N)(Y,N,Y)(Y,N,N)(N,Y,Y)(N,Y,N)(N,N,Y)(N,N,N)ValueofN322121104.x=0,1,2,...,12.5.a.S={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)}b.ExperimentalOutcome(1,1)(1,2)(1,3)(2,1)(2,2)(2,3)NumberofStepsRequired2343456.a.values:0,1,2,...,20discreteb.values:0,1,2,...discretec.values:0,1,2,...,50discreted.values:0x8continuouse.values:x>0continuous7.a.f(x)0forallvaluesofx.f(x)=1Therefore,itisaproperprobabilitydistribution.b.Probabilityx=30isf(30)=.25c.Probabilityx25isf(20)+f(25)=.20+.15=.35d.Probabilityx>30isf(35)=.4013-89 8.a.xf(x)13/20=.1525/20=.2538/20=.4044/20=.20Total1.00b.f(x).4.3.2.1x1234c.f(x)0forx=1,2,3,4.f(x)=19.a.xf(x)115/462=0.032232/462=0.069384/462=0.1824300/462=0.650531/462=0.067b.13-90 f(x)0.600.450.300.15x012345c.Allf(x)0f(x)=0.032+0.069+0.182+0.650+0.067=1.00010.a.xf(x)10.0520.0930.0340.4250.411.00b.xf(x)10.0420.1030.1240.4650.281.00c.P(4or5)=f(4)+f(5)=0.42+0.41=0.83d.Probabilityofverysatisfied:0.28e.Seniorexecutivesappeartobemoresatisfiedthanmiddlemanagers.83%ofseniorexecutiveshaveascoreof4or5with41%reportinga5.Only28%ofmiddlemanagersreportbeingverysatisfied.11.a.DurationofCallxf(x)10.2520.2530.2540.251.0013-91 b.f(x)0.300.200.10x01234c.f(x)0andf(1)+f(2)+f(3)+f(4)=0.25+0.25+0.25+0.25=1.00d.f(3)=0.25e.P(overtime)=f(3)+f(4)=0.25+0.25=0.5012.a.Yes;f(x)0forallxandf(x)=.15+.20+.30+.25+.10=1b.P(1200orless)=f(1000)+f(1100)+f(1200)=.15+.20+.30=.6513.a.Yes,sincef(x)0forx=1,2,3andf(x)=f(1)+f(2)+f(3)=1/6+2/6+3/6=1b.f(2)=2/6=.333c.f(2)+f(3)=2/6+3/6=.83314.a.f(200)=1-f(-100)-f(0)-f(50)-f(100)-f(150)=1-.95=.05ThisistheprobabilityMRAwillhavea$200,000profit.b.P(Profit)=f(50)+f(100)+f(150)+f(200)=.30+.25+.10+.05=.70c.P(atleast100)=f(100)+f(150)+f(200)=.25+.10+.05=.4015.a.xf(x)xf(x)3.25.756.503.009.252.251.006.00E(x)==6.0013-92 b.xx-(x-)2f(x)(x-)2f(x)3-39.252.25600.500.00939.252.254.502Var(x)==4.50c.=4.50=2.1216.a.yf(y)yf(y)2.20.404.301.207.402.808.10.801.005.20E(y)==5.20b.yy-(y-)2f(y)(y-)2f(y)2-3.2010.24.202.0484-1.201.44.30.43271.803.24.401.29682.807.84.10.7844.560Var()y456.456..21417.a/b.xf(x)xf(x)x-(x-)22(x-)f(x)0.10.00-2.456.0025.6002501.15.15-1.452.1025.3153752.30.60-.45.2025.0607503.20.60.55.3025.0605004.15.601.552.4025.3603755.10.502.556.5025.6502502.452.047500E(x)==2.452=2.0475=1.430918.a/b.xf(x)xf(x)x-(x-)22(x-)f(x)0.010-2.35.290.05291.23.23-1.31.690.38872.41.82-0.30.090.03693.20.600.70.490.0984.10.401.72.890.2895.05.252.77.290.364513-93 2.31.23E(x)=2.3Var(x)=1.231.231.11Theexpectedvalue,E(x)=2.3,oftheprobabilitydistributionisthesameasthatreportedinthe1997StatisticalAbstractoftheUnitedStates.19.a.E(x)=xf(x)=0(.50)+2(.50)=1.00b.E(x)=xf(x)=0(.61)+3(.39)=1.17c.Theexpectedvalueofa3-pointshotishigher.So,iftheseprobabilitiesholdup,theteamwillmakemorepointsinthelongrunwiththe3-pointshot.20.a.xf(x)xf(x)0.900.00400.0416.001000.0330.002000.0120.004000.0140.006000.0160.001.00166.00E(x)=166.Ifthecompanychargedapremiumof$166.00theywouldbreakeven.b.GaintoPolicyHolderf(Gain)(Gain)f(Gain)-260.00.90-234.00140.00.045.60740.00.0322.201,740.00.0117.403,740.00.0137.405,740.00.0157.40-94.00E(gain)=-94.00.Thepolicyholderismoreconcernedthatthebigaccidentwillbreakhimthanwiththeexpectedannuallossof$94.00.21.a.E(x)=xf(x)=0.05(1)+0.09(2)+0.03(3)+0.42(4)+0.41(5)=4.05b.E(x)=xf(x)=0.04(1)+0.10(2)+0.12(3)+0.46(4)+0.28(5)=3.8422c.Executives:=(x-)f(x)=1.247522MiddleManagers:=(x-)f(x)=1.1344d.Executives:=1.1169MiddleManagers:=1.065113-94 e.Theseniorexecutiveshaveahigheraveragescore:4.05vs.3.84forthemiddlemanagers.Theexecutivesalsohaveaslightlyhigherstandarddeviation.22.a.E(x)=xf(x)=300(.20)+400(.30)+500(.35)+600(.15)=445Themonthlyorderquantityshouldbe445units.b.Cost:445@$50=$22,250Revenue:300@$70=21,000$1,250Loss23.a.Laptop:E(x)=.47(0)+.45(1)+.06(2)+.02(3)=.63Desktop:E(x)=.06(0)+.56(1)+.28(2)+.10(3)=1.422222b.Laptop:Var(x)=.47(-.63)+.45(.37)+.06(1.37)+.02(2.37)=.47312222Desktop:Var(x)=.06(-1.42)+.56(-.42)+.28(.58)+.10(1.58)=.5636c.Fromtheexpectedvaluesinpart(a),itisclearthatthetypicalsubscriberhasmoredesktopcomputersthanlaptops.Thereisnotmuchdifferenceinthevariancesforthetwotypesofcomputers.24.a.MediumE(x)=xf(x)=50(.20)+150(.50)+200(.30)=145Large:E(x)=xf(x)=0(.20)+100(.50)+300(.30)=140Mediumpreferred.b.Mediumxf(x)x-(x-)22(x-)f(x)50.20-9590251805.0150.5052512.5200.30553025907.52=2725.0Largeyf(y)y-(y-)22(y-)f(y)0.20-140196003920100.50-401600800300.301602560076802=12,400Mediumpreferredduetolessvariance.13-95 25.a.SSFFSF22!11b.f(1)(.4)(.6)(.4)(.6).4811!1!22!02c.f(0)(.4)(.6)(1)(.36).3600!2!22!20d.f(2)(.4)(.6)(.16)(1).1622!0!e.P(x1)=f(1)+f(2)=.48+.16=.64f.E(x)=np=2(.4)=.8Var(x)=np(1-p)=2(.4)(.6)=.48=.48=.692826.a.f(0)=.3487b.f(2)=.1937c.P(x2)=f(0)+f(1)+f(2)=.3487+.3874+.1937=.9298d.P(x1)=1-f(0)=1-.3487=.6513e.E(x)=np=10(.1)=1f.Var(x)=np(1-p)=10(.1)(.9)=.9=.9=.948727.a.f(12)=.1144b.f(16)=.130413-96 c.P(x16)=f(16)+f(17)+f(18)+f(19)+f(20)=.1304+.0716+.0278+.0068+.0008=.2374d.P(x15)=1-P(x16)=1-.2374=.7626e.E(x)=np=20(.7)=14f.Var(x)=np(1-p)=20(.7)(.3)=4.2=4.2=2.049462428.a.f(2)(.33)(.67).32922b.P(atleast2)=1-f(0)-f(1)660615=1(.33)(.67)(.33)(.67)01=1-.0905-.2673=.642210010c.f(10)(.33)(.67).0182029.P(AtLeast5)=1-f(0)-f(1)-f(2)-f(3)-f(4)=1-.0000-.0005-.0031-.0123-.0350=.949130.a.Probabilityofadefectivepartbeingproducedmustbe.03foreachtrial;trialsmustbeindependent.b.Let:D=defectiveG=notdefectiveExperimentalNumber1stpart2ndpartOutcomeDefectiveD(D,D)2DG(D,G)1.GD(G,D)1G(G,G)0c.2outcomesresultinexactlyonedefect.d.P(nodefects)=(.97)(.97)=.9409P(1defect)=2(.03)(.97)=.058213-97 P(2defects)=(.03)(.03)=.000931.Binomialn=10andp=.0510!xx10fx()(.)(.)0595xx!(10)!a.Yes.Sincetheyareselectedrandomly,pisthesamefromtrialtotrialandthetrialsareindependent.b.f(2)=.0746c.f(0)=.5987d.P(Atleast1)=1-f(0)=1-.5987=.401332.a..90b.P(atleast1)=f(1)+f(2)2!11f(1)=(.9)(.1)1!1!=2(.9)(.1)=.182!20f(2)=(.9)(.1)2!0!=1(.81)(1)=.81P(atleast1)=.18+.81=.99AlternativelyP(atleast1)=1-f(0)2!02f(0)=(.9)(.1)=.010!2!Therefore,P(atleast1)=1-.01=.99c.P(atleast1)=1-f(0)3!03f(0)=(.9)(.1)=.0010!3!Therefore,P(atleast1)=1-.001=.999d.Yes;P(atleast1)becomesverycloseto1withmultiplesystemsandtheinabilitytodetectanattackwouldbecatastrophic.33.a.Usingthebinomialformulaorthetableofbinomialprobabilitieswithp=.5andn=20,wefind:xf(x)120.1201130.0739140.0370150.0148160.004613-98 170.0011180.0002190.0000200.00000.2517Theprobability12ormorewillsendrepresentativesis0.2517.b.Usingthebinomialformulaorthetables,wefind:xf(x)00.000010.000020.000230.001140.004650.01480.0207c.E(x)=np=20(0.5)=102d.=np(1-p)=20(0.5)(0.5)=5=5=2.236134.a.f(3)=.0634(fromtables)b.Theanswerhereisthesameaspart(a).Theprobabilityof12failureswithp=.60isthesameastheprobabilityof3successeswithp=.40.c.f(3)+f(4)+···+f(15)=1-f(0)-f(1)-f(2)=1-.0005-.0047-.0219=.972935.a.f(0)+f(1)+f(2)=.0115+.0576+.1369=.2060b.f(4)=.2182c.1-[f(0)+f(1)+f(2)+f(3)]=1-.2060-.2054=.5886d.=np=20(.20)=436.xf(x)x-(x-)22(x-)f(x)0.343-.9.81.277831.441.1.01.004412.1891.11.21.228693.0272.14.41.1190721.000=.6300037.E(x)=np=30(0.29)=8.72=np(1-p)=30(0.29)(0.71)=6.17713-99 =6.177=2.485x33e38.a.fx()x!233e9(.0498)b.f(2).22412!2133ec.f(1)3(.0498).14941!d.P(x2)=1-f(0)-f(1)=1-.0498-.1494=.8008x22e39.a.fx()x!b.=6for3timeperiodsx66ec.fx()x!222e4(.1353)d.f(2).27062!2666ee.f(6).16066!544ef.f(5).15635!40.a.=48(5/60)=43-44e(64)(.0183)f(3)===.19523!6b.=48(15/60)=1210-1212ef(10)==.104810!c.=48(5/60)=4Iexpect4callerstobewaitingafter5minutes.0-44ef(0)==.01830!Theprobabilitynonewillbewaitingafter5minutesis.0183.d.=48(3/60)=2.413-100 0-2.42.4ef(0)==.09070!Theprobabilityofnointerruptionsin3minutesis.0907.41.a.30perhourb.=1(5/2)=5/23(5/2)(5/2)ef(3).21383!0(5/2)(5/2)e(5/2)c.fe(0).08210!e42.a.fx()x!244e16(0.0183)f(2)8(0.0183)0.14652!2b.Fora3-monthperiod:=1c.Fora6-monthperiod:022e2fe(0)0.13530!Theprobabilityof1ormoreflights=1-f(0)=1-0.1353=0.864701010e1043.a.fe(0).0000450!b.f(0)+f(1)+f(2)+f(3)f(0)=.000045(parta)11010ef(1).000451!Similarly,f(2)=.00225,f(3)=.0075andf(0)+f(1)+f(2)+f(3)=.010245c.2.5arrivals/15sec.periodUse=2.502.52.5ef(0).08210!13-101 d.1-f(0)=1-.0821=.917944.Poissondistributionappliesa.=1.25permonth01.251.25eb.f(0)0.28650!11.251.25ec.f(1)0.35811!d.P(Morethan1)=1-f(0)-f(1)=1-0.2865-0.3581=0.355445.a.For1week,=450/52=8.6518.658.65e8.65b.fe(0)0.00020!c.Fora1-dayperiod:=450/365=1.230–1.231.23e–1.23f(0)==e=0.29230!1–1.231.23ef(1)==1.23(0.2923)=0.35951Probabilityof2ormoredeaths=1-f(0)-f(1)=1-0.2923-0.3595=0.348231033!7!1411!2!3!4!(3)(35)46.a.f(1).501010!21044!6!3103222(3)(1)b.f(2).067104523103020(1)(21)c.f(0).46671045213-102 3103242(3)(21)d.f(2).3010210441543103(4)(330)47.f(3).43961530031048.HypergeometricwithN=10andr=66421(15)(4)a.f(2).50101203b.Mustbe0or1preferCokeClassic.6412(6)(6)f(1).301012036403(1)(4)f(0).0333101203P(MajorityPepsi)=f(1)+f(0)=.333349.Partsa,b&cinvolvethehypergeometricdistributionwithN=52andn=2a.r=20,x=2203220(190)(1)f(2).14335213262b.r=4,x=244820(6)(1)f(2).0045521326213-103 c.r=16,x=2163620(120)(1)f(2).09055213262d.Part(a)providestheprobabilityofblackjackplustheprobabilityof2acesplustheprobabilityoftwo10s.Tofindtheprobabilityofblackjackwesubtracttheprobabilitiesin(b)and(c)fromtheprobabilityin(a).P(blackjack)=.1433-.0045-.0905=.048350.N=60n=10a.r=20x=0F20IF40IF40!Ibg1HG0KJHG10KJHG10!30!KJF40!IF10!50!If(0)=F60I60!HG10!30!KJHG60!KJHG10KJ10!50!40393837363534333231=60595857565554535251.01b.r=20x=1F20IF40IHG1KJHG9KJF40!IF10!50!If(0)=20KJF60IHG931!!KJHG60!HG10KJ.07c.1-f(0)-f(1)=1-.08=.92d.SameastheprobabilityonewillbefromHawaii.Inpartbthatwasfoundtoequalapproximately.07.111423(55)(364)51.a.f(2).37682553,130513-104 141123(91)(165)b.f(2).28262553,1305141150(2002)(1)c.f(5).03772553,1305141105(1)(462)d.f(0).00872553,130552.HypergeometricwithN=10andr=2.Focusontheprobabilityof0defectives,thentheprobabilityofrejectingtheshipmentis1-f(0).a.n=3,x=0280356f(0).4667101203P(Reject)=1-.4667=.5333b.n=4,x=0280470f(0).3333102104P(Reject)=1-.3333=.6667c.n=5,x=0280556f(0).2222102525P(Reject)=1-.2222=.7778d.Continuetheprocess.n=7wouldberequiredwiththeprobabilityofrejecting=.933313-105 53.a.,b.andc.xf(x)xf(x)x-(x-)22(x-)f(x)10.180.18-2.305.290.952220.180.36-1.301.690.608430.030.09-0.300.090.008140.381.520.700.490.744850.231.151.702.893.32351.003.305.63702E(x)==3.30=5.6370=5.6370=2.374254.a.andb.xf(x)xf(x)x-(x-)22(x-)f(x)10.020.02-2.646.96960.13939220.060.12-1.642.68960.16137630.280.84-0.640.40960.11468840.542.160.360.12960.06998450.100.501.361.84960.1849601.003.640.670400f(x)0andf(x)=1E(x)==3.642Var(x)==0.6704c.Peopledoappeartobelievethestockmarketisovervalued.Theaverageresponseisslightlyoverhalfwaybetween“fairlyvalued”and“somewhatovervalued.”55.a.xf(x)9.3010.2011.2512.0513.20b.E(x)=xf(x)=9(.30)+10(.20)+11(.25)+12(.05)+13(.20)=10.65Expectedvalueofexpenses:$10.65million2c.Var(x)=(x-)f(x)222=(9-10.65)(.30)+(10-10.65)(.20)+(11-10.65)(.25)22+(12-10.65)(.05)+(13-10.65)(.20)13-106 =2.1275d.LooksGood:E(Profit)=12-10.65=1.35millionHowever,thereisa.20probabilitythatexpenseswillequal$13millionandthecollegewillrunadeficit.56.a.n=20andx=320317f(3)(0.04)(0.04)0.03643b.n=20andx=020020f(0)(0.04)(0.96)0.44200c.E(x)=np=1200(0.04)=48Theexpectednumberofappealsis48.d.=np(1-p)=1200(0.04)(0.96)=46.08=46.08=6.788257.a.WemusthaveE(x)=np10Withp=.4,thisleadsto:n(.4)10n25b.Withp=.12,thisleadsto:n(.12)10n83.33So,wemustcontact84peopleinthisagegrouptohaveanexpectednumberofinternetusersofatleast10.c.25(.4)(.6)2.45d.25(.12)(.88)1.6258.Sincetheshipmentislargewecanassumethattheprobabilitiesdonotchangefromtrialtotrialandusethebinomialprobabilitydistribution.a.n=5505f(0)(0.01)(0.99)0.9510013-107 514b.f(1)(0.01)(0.99)0.04801c.1-f(0)=1-.9510=.0490d.No,theprobabilityoffindingoneormoreitemsinthesampledefectivewhenonly1%oftheitemsinthepopulationaredefectiveissmall(only.0490).Iwouldconsideritlikelythatmorethan1%oftheitemsaredefective.59.a.E(x)=np=100(.041)=4.1b.Var(x)=np(1-p)=100(.041)(.959)=3.93193.93191.982960.a.E(x)=800(.41)=328b.np(1p)800(.41)(.59)13.91c.Forthisonep=.59and(1-p)=.41,buttheansweristhesameasinpart(b).Forabinomialprobabilitydistribution,thevarianceforthenumberofsuccessesisthesameasthevarianceforthenumberoffailures.Ofcourse,thisalsoholdstrueforthestandarddeviation.61.=15probof20ormorearrivals=f(20)+f(21)+···=.0418+.0299+.0204+.0133+.0083+.0050+.0029+.0016+.0009+.0004+.0002+.0001+.0001=.124962.=1.5probof3ormorebreakdownsis1-[f(0)+f(1)+f(2)].1-[f(0)+f(1)+f(2)]=1-[.2231+.3347+.2510]=1-.8088=.191263.=10f(4)=.0189333e64.a.f()302240.3!b.f(3)+f(4)+···=1-[f(0)+f(1)+f(2)]0-33e-3f(0)==e=.04980!13-108 Similarly,f(1)=.1494,f(2)=.22401-[.0498+.1494+.2241]=.576765.HypergeometricN=52,n=5andr=4.F4IF48IHG2KJHG3KJ617296()a..0399F52I2598960,,HG5KJF4IF48IHG1KJHG4KJ4194580()b..2995F52I2598960,,HG5KJF4IF48IHG0KJHG5KJ1712304,,c..6588F52I2598960,,HG5KJd.1-f(0)=1-.6588=.341266.UsetheHypergeometricprobabilitydistributionwithN=10,n=2,andr=4.F4IF6IHG1KJHG1KJ()()46a.f()1.5333F10I45HG2KJF4IF6IHG2KJHG0KJ()()61b.f()2.1333F10I45HG2KJF4IF6IHG0KJHG2KJ()()115c.f()0.3333F10I45HG2KJChapter6ContinuousProbabilityDistributions13-109 LearningObjectives1.Understandthedifferencebetweenhowprobabilitiesarecomputedfordiscreteandcontinuousrandomvariables.2.Knowhowtocomputeprobabilityvaluesforacontinuousuniformprobabilitydistributionandbeabletocomputetheexpectedvalueandvarianceforsuchadistribution.3.Beabletocomputeprobabilitiesusinganormalprobabilitydistribution.Understandtheroleofthestandardnormaldistributioninthisprocess.4.Beabletocomputeprobabilitiesusinganexponentialprobabilitydistribution.5.UnderstandtherelationshipbetweenthePoissonandexponentialprobabilitydistributions.Solutions:1.a.13-110 f(x)321x.501.01.52.0b.P(x=1.25)=0.Theprobabilityofanysinglepointiszerosincetheareaunderthecurveaboveanysinglepointiszero.c.P(1.0x1.25)=2(.25)=.50d.P(1.20135)=(1/20)(140-135)=0.25120140d.Ex()130minutes24.a.f(x)1.51.0.5x0123b.P(.25.60)=1(.40)=.405.a.LengthofInterval=261.2-238.9=22.31for238.9x261.2fx()22.30elsewhereb.Note:1/22.3=0.045P(x<250)=(0.045)(250-238.9)=0.4995Almosthalfdrivetheballlessthan250yards.c.P(x255)=(0.045)(261.2-255)=0.279d.P(245x260)=(0.045)(260-245)=0.67513-112 e.P(x250)=1-P(x<250)=1-0.4995=0.5005Theprobabilityofanyonedrivingit250yardsormoreis0.5005.With60players,theexpectednumberdrivingit250yardsormoreis(60)(0.5005)=30.03.Rounding,Iwouldexpect30ofthesewomentodrivetheball250yardsormore.6.a.P(12x12.05)=.05(8)=.40b.P(x12.02)=.08(8)=.64c.(Px11.98)Px(12.02).005(8).04.64.08(8)Therefore,theprobabilityis.04+.64=.687.a.P(10,000x<12,000)=2000(1/5000)=.40Theprobabilityyourcompetitorwillbidlowerthanyou,andyougetthebid,is.40.b.P(10,000x<14,000)=4000(1/5000)=.80c.Abidof$15,000givesaprobabilityof1ofgettingtheproperty.d.Yes,thebidthatmaximizesexpectedprofitis$13,000.Theprobabilityofgettingthepropertywithabidof$13,000isP(10,000x<13,000)=3000(1/5000)=.60.Theprobabilityofnotgettingthepropertywithabidof$13,000is.40.Theprofityouwillmakeifyougetthepropertywithabidof$13,000is$3000=$16,000-13,000.Soyourexpectedprofitwithabidof$13,000isEP($13,000)=.6($3000)+.4(0)=$1800.Ifyoubid$15,000theprobabilityofgettingthebidis1,buttheprofitifyoudogetthebidisonly$1000=$16,000-15,000.Soyourexpectedprofitwithabidof$15,000isEP($15,000)=1($1000)+0(0)=$1,000.13-113 8.=107080901001101201309.a.=535404550556065b..6826since45and55arewithinplusorminus1standarddeviationfromthemeanof50.c..9544since40and60arewithinplusorminus2standarddeviationsfromthemeanof50.10.-3-2-10+1+2+3a..3413b..4332c..4772d..493813-114 11.a..3413Theseprobabilityvaluesarereaddirectlyfromthetableofareasforthestandardb..4332normalprobabilitydistribution.SeeTable1inAppendixB.c..4772d..4938e..498612.a..2967b..4418c..5000-.1700=.3300d..0910+.5000=.5910e..3849+.5000=.8849f..5000-.2612=.238813.a..6879-.0239=.6640b..8888-.6985=.1903c..9599-.8508=.109114.a.Usingthetableofareasforthestandardnormalprobabilitydistribution,theareaof.4750correspondstoz=1.96.b.Usingthetable,theareaof.2291correspondstoz=.61.c.Lookinthetableforanareaof.5000-.1314=.3686.Thisprovidesz=1.12.d.Lookinthetableforanareaof.6700-.5000=.1700.Thisprovidesz=.44.15.a.Lookinthetableforanareaof.5000-.2119=.2881.Sincethevalueweareseekingisbelowthemean,thezvaluemustbenegative.Thus,foranareaof.2881,z=-.80.b.Lookinthetableforanareaof.9030/2=.4515;z=1.66.c.Lookinthetableforanareaof.2052/2=.1026;z=.26.d.Lookinthetableforanareaof.4948;z=2.56.e.Lookinthetableforanareaof.1915.Sincethevalueweareseekingisbelowthemean,thezvaluemustbenegative.Thus,z=-.50.16.a.Lookinthetableforanareaof.5000-.0100=.4900.Theareavalueinthetableclosestto.4900providesthevaluez=2.33.b.Lookinthetableforanareaof.5000-.0250=.4750.Thiscorrespondstoz=1.96.13-115 c.Lookinthetableforanareaof.5000-.0500=.4500.Since.4500isexactlyhalfwaybetween.4495(z=1.64)and.4505(z=1.65),weselectz=1.645.However,z=1.64orz=1.65arealsoacceptableanswers.d.Lookinthetableforanareaof.5000-.1000=.4000.Theareavalueinthetableclosestto.4000providesthevaluez=1.28.17.Convertmeantoinches:=69a.Atx=72z=72-69=13P(x72)=0.5000+0.3413=0.8413P(x>72)=1-0.8413=0.1587b.Atx=60z=60-69=–33P(x60)=0.5000+0.4986=0.9986P(x<60)=1-0.9986=0.0014c.Atx=70z=70-69=0.333P(x70)=0.5000+0.1293=0.6293Atx=66z=66-69=–13P(x66)=0.5000-0.3413=0.1587P(66x70)=P(x70)-P(x66)=0.6293-0.1587=0.4706d.P(x72)=1-P(x>72)=1-0.1587=0.841318.a.FindP(x60)Atx=60z=60-49=0.6916P(x<60)=0.5000+0.2549=0.7549P(x60)=1-P(x<60)=0.2451b.FindP(x30)Atx=30z=30-49=–1.1916P(x30)=0.5000-0.3830=0.1170c.Findz-scoresothatP(zz-score)=0.10z-score=1.28cutsoff10%inuppertail13-116 Now,solveforcorrespondingvalueofx.x49128.16x=49+(16)(1.28)=69.48So,10%ofsubscribersspend69.48minutesormorereadingTheWallStreetJournal.19.Wehave=3.5and=.8.5.03.5a.z1.88.8P(x>5.0)=P(z>1.88)=1-P(z<1.88)=1-.9699=.0301Therainfallexceeds5inchesin3.01%oftheAprils.33.5b.z.63.8P(x<3.0)=P(z<-.63)=P(z>.63)=1-P(z<.63)=1-.7357=.2643Therainfallislessthan3inchesin26.43%oftheAprils.c.z=1.28cutsoffapproximately.10intheuppertailofanormaldistribution.x=3.5+1.28(.8)=4.524Ifitrains4.524inchesormore,Aprilwillbeclassifiedasextremelywet.20.Weuse=27and=81127a.z28P(x11)=P(z-2)=.5000-.4772=.0228Theprobabilityarandomlyselectedsubscriberspendslessthan11hoursonthecomputeris.025.4027b.z1.638P(x>40)=P(z>1.63)=1-P(z1.63)=1-.9484=.05165.16%ofsubscribersspendover40hoursperweekusingthecomputer.c.Az-valueof.84cutsoffanareaof.20intheuppertail.x=27+.84(8)=33.72Asubscriberwhousesthecomputer33.72hoursormorewouldbeclassifiedasaheavyuser.13-117 21.Fromthenormalprobabilitytables,az-valueof2.05cutsoffanareaofapproximately.02intheuppertailofthedistribution.x=+z=100+2.05(15)=130.75Ascoreof131orbettershouldqualifyapersonformembershipinMensa.22.Use=441.84and=90a.At400400441.84z.4690At500500441.84z.6590P(0z<.65)=.2422P(-.46z<0)=.1772P(400z500)=.1772+.2422=.4194Theprobabilityaworkerearnsbetween$400and$500is.4194.b.Mustfindthez-valuethatcutsoffanareaof.20intheuppertail.Usingthenormaltables,wefindz=.84cutsoffapproximately.20intheuppertail.So,x=+z=441.84+.84(90)=517.44Weeklyearningsof$517.44orabovewillputaproductionworkerinthetop20%.250441.84c.At250,z2.1390P(x250)=P(z-2.13)=.5000-.4834=.0166Theprobabilityarandomlyselectedproductionworkerearnslessthan$250perweekis.0166.608023.a.z2Areatoleftis.5000-.4772=.022810b.Atx=606080z2Areatoleftis.022810Atx=757580z.5Areatoleftis.308510P(60x75)=.3085-.0228=.285713-118 9080c.z1Area=.5000-.3413=.158710Therefore15.87%ofstudentswillnotcompleteontime.(60)(.1587)=9.522Wewouldexpect9.522studentstobeunabletocompletetheexamintime.xi24.a.x902.75n2()xxis114.185n1Wewillusexasanestimateofandsasanestimateofinparts(b)-(d)below.b.Rememberthedataareinthousandsofshares.At800800902.75z.90114.185P(x800)=P(z-.90)=1-P(z.90)=1-.8159=.1841Theprobabilitytradingvolumewillbelessthan800millionsharesis.1841c.At10001000902.75z.85114.185P(x1000)=P(z.85)=1-P(z.85)=1-.8023=.1977Theprobabilitytradingvolumewillexceed1billionsharesis.1977d.Az-valueof1.645cutsoffanareaof.05intheuppertailx=+z=902.75+1.645(114.185)=1,090.584Theyshouldissueapressreleaseanytimesharevolumeexceeds1,091million.25.a.FindP(x>100)Atx=100z=100-110=–0.520P(x>100)=P(z.5)=0.6915b.FindP(x90)Atx=9013-119 z=90-110=–120P(x90)=.5000-.3413=0.1587c.FindP(80x130)Atx=130z=130-110=120P(x130)=0.8413Atx=8080110z1.5Areatoleftis.066820P(80x130)=.8413-.0668=.7745-6/826.a.P(x6)=1-e=1-.4724=.5276-4/8b.P(x4)=1-e=1-.6065=.3935c.P(x6)=1-P(x6)=1-.5276=.4724d.P(4x6)=P(x6)-P(x4)=.5276-.3935=.1341Pxx()1ex0/327.a.0-2/3b.P(x2)=1-e=1-.5134=.486633/-1c.P(x3)=1-P(x3)=1-(1-e)=e=.3679-5/3d.P(x5)=1-e=1-.1889=.8111e.P(2x5)=P(x5)-P(x2)=.8111-.4866=.3245-10/2028.a.P(x10)=1-e=.3935b.P(x30)=1-P(x30)=1-(1-e-30/20)=e-30/20=.2231c.P(10x30)=P(x30)-P(x10)=(1-e-30/20)-(1-e-10/20)=e-10/20-e-30/20=.6065-.2231=.383413-120 29.a.f(x).09.08.07.06.05.04.03.02.01x6121824-12/12b.P(x12)=1-e=1-.3679=.6321-6/12c.P(x6)=1-e=1-.6065=.3935d.P(x30)=1-P(x<30)-30/12=1-(1-e)=.082130.a.50hours-25/50b.P(x25)=1-e=1-.6065=.3935-100/50c.P(x100)=1-(1-e)=.1353-2/2.7831.a.P(x2)=1-e=.5130-5/2.78-5/2.78b.P(x5)=1-P(x5)=1-(1-e)=e=.1655-2.78/2.78-1c.P(x2.78)=1-P(x2.78)=1-(1-e)=e=.3679Thismayseemsurprisingsincethemeanis2.78minutes.But,fortheexponentialdistribution,theprobabilityofavaluegreaterthanthemeanissignificantlylessthantheprobabilityofavaluelessthanthemean.13-121 32.a.IftheaveragenumberoftransactionsperyearfollowsthePoissondistribution,thetimebetweentransactionsfollowstheexponentialdistribution.So,1=ofayear3011and301/30-30xthenf(x)=30eb.Amonthis1/12ofayearso,1130/1230/12Px1Px1(1e)e.08211212TheprobabilityofnotransactionduringJanuaryisthesameastheprobabilityofnotransactionduringanymonth:.0821c.Since1/2monthis1/24ofayear,wecompute,130/24Px1e1.2865.71352433.a.Letx=salesprice($1000s)1for200x225fx()250elsewhereb.P(x215)=(1/25)(225-215)=0.40c.P(x<210)=(1/25)(210-200)=0.40d.E(x)=(200+225)/2=212,500Ifshewaits,herexpectedsalepricewillbe$2,500higherthanifshesellsitbacktohercompanynow.However,thereisa0.40probabilitythatshewillgetless.It’saclosecall.But,theexpectedvalueapproachtodecisionmakingwouldsuggestsheshouldwait.34.a.Foranormaldistribution,themeanandthemedianareequal.63,000b.Findthez-scorethatcutsoff10%inthelowertail.z-score=-1.28Solvingforx,–1.28=x–63,00015,00013-122 x=63,000-1.28(15000)=43,800Thelower10%ofmortgagedebtis$43,800orless.c.FindP(x>80,000)Atx=80,000z=80,000–63,000=1.1315,000P(x>80,000)=1.0000-.8708=0.1292d.Findthez-scorethatcutsoff5%intheuppertail.z-score=1.645.Solveforx.1.645=x–63,00015,000x=63,000+1.645(15,000)=87,675Theupper5%ofmortgagedebtisinexcessof$87,675.35.a.P(defect)=1-P(9.85x10.15)=1-P(-1z1)=1-.6826=.3174Expectednumberofdefects=1000(.3174)=317.4b.P(defect)=1-P(9.85x10.15)=1-P(-3z3)=1-.9972=.0028Expectednumberofdefects=1000(.0028)=2.8c.Reducingtheprocessstandarddeviationcausesasubstantialreductioninthenumberofdefects.36.a.At11%,z=-1.2313-123 –1.23=x–1800–2071=1800–2071Therefore,==$220.33–1.2320002071b.z.32Areatoleftis.5000-.3255=.3745220.3325002071z1.95Areatoleftis.9744220.33P(2000x2500)=.9744-.3745=.5999c.z=-1.88x=2071-1.88(220.33)=$1656.7837.=10,000=1500a.Atx=12,00012,00010,000z1.33Areatoleftis.90821500P(x>12,000)=1.0000-.9082=.0918b.At.95x-10,000z=1.645=1500Therefore,x=10,000+1.645(1500)=12,468.95%0.0510,00012,46812,468tubesshouldbeproduced.38.a.Atx=200200150z2Area=.477225P(x>200)=.5-.4772=.0228b.ExpectedProfit=ExpectedRevenue-ExpectedCost13-124 =200-150=$5039.a.FindP(80,000x150,000)Atx=150,000z=150,000–126,681=0.7830,000P(x150,000)=0.7823Atx=80,000z=80,000–126,681=–1.5630,000P(x80,000)=.5000-.4406=0.0594P(80,000x150,000)=0.7823-0.0594=0.7229b.FindP(x<50,000)Atx=50,000z=50,000–126,681=–2.5630,000P(x<50,000)=.5000-.4948=0.0052c.Findthez-scorecuttingoff95%inthelefttail.z-score=1.645.Solveforx.1.645=x–126,68130,000x=126,681+1.645(30,000)=176,031Theprobabilityis0.95thatthenumberoflostjobswillnotexceed176,031.40.a.At400,400450z.500100Areatoleftis.3085At500,500450z.500100Areatoleftis.6915P(400x500)=.6915-.3085=.383038.3%willscorebetween400and500.b.At630,13-125 630450z1.8010096.41%doworseand3.59%dobetter.c.At480,480450z.30100Areatoleftis.617938.21%areacceptable.41.a.At75,00075,00067,000z1.147,000P(x>75,000)=P(z>1.14)=1-P(z1.14)=1-.8729=.1271Theprobabilityofawomanreceivingasalaryinexcessof$75,000is.1271b.At75,00075,00065,500z1.367,000P(x>75,000)=P(z>1.36)=1-P(z1.36)=1-.9131=.0869Theprobabilityofamanreceivingasalaryinexcessof$75,000is.0869c.Atx=50,00050,00067,000z2.437,000P(x<50,000)=P(z<-2.43)=1-P(z<2.43)=1-.9925=.0075Theprobabilityofawomanreceivingasalarybelow$50,000isverysmall:.0075d.Theanswertothisisthemalecopywritersalarythatcutsoffanareaof.01intheuppertailofthedistributionformalecopywriters.Usez=2.33x=65,500+2.33(7,000)=81,810Awomanwhomakes$81,810ormorewillearnmorethan99%ofhermalecounterparts.42.=.6At2%z=-2.05x=18x18z2.05.613-126 =18+2.05(.6)=19.23oz.0.0218=19.23Themeanfillingweightmustbe19.23oz.-15/3643.a.P(x15)=1-e=1-.6592=.3408-45/36b.P(x45)=1-e=1-.2865=.7135ThereforeP(15x45)=.7135-.3408=.3727c.P(x60)=1-P(x<60)-60/36=1-(1-e)=.188944.a.4hours-x/4b.f(x)=(1/4)eforx0-1/4c.P(x1)=1-P(x<1)=1-(1-e)=.7788-8/4d.P(x>8)=1-P(x8)=e=.13531x/1.245.a.fx()eforx01.2-1/1.2-.5/1.2b.P(.5x1.0)=P(x1.0)-P(x.5)=(1-e)-(1-e)=.5654-.3408=.2246c.P(x>1)=1-P(x1)=1-.5654=.4346146.a.05.therefore=2minutes=meantimebetweentelephonecallsb.Note:30seconds=.5minutes-.5/2P(x.5)=1-e=1-.7788=.2212-1/2c.P(x1)=1-e=1-.6065=.3935-5/2d.P(x5)=1-P(x<5)=1-(1-e)=.082113-127 Chapter7SamplingandSamplingDistributionsLearningObjectives1.Understandtheimportanceofsamplingandhowresultsfromsamplescanbeusedtoprovideestimatesofpopulationcharacteristicssuchasthepopulationmean,thepopulationstandarddeviationand/orthepopulationproportion.2.Knowwhatsimplerandomsamplingisandhowsimplerandomsamplesareselected.3.Understandtheconceptofasamplingdistribution.4.Knowthecentrallimittheoremandtheimportantroleitplaysinsampling.5.Specificallyknowthecharacteristicsofthesamplingdistributionofthesamplemean(x)andthesamplingdistributionofthesampleproportion(p).6.Becomeawareofthepropertiesofpointestimatorsincludingunbiasedness,consistency,andefficiency.7.Learnaboutavarietyofsamplingmethodsincludingstratifiedrandomsampling,clustersampling,systematicsampling,conveniencesamplingandjudgmentsampling.8.Knowthedefinitionofthefollowingterms:simplerandomsamplingfinitepopulationcorrectionfactorsamplingwithreplacementstandarderrorsamplingwithoutreplacementunbiasednesssamplingdistributionconsistencypointestimatorefficiency13-128 Solutions:1.a.AB,AC,AD,AE,BC,BD,BE,CD,CE,DEb.With10samples,eachhasa1/10probability.c.EandCbecause8and0donotapply.;5identifiesE;7doesnotapply;5isskippedsinceEisalreadyinthesample;3identifiesC;2isnotneededsincethesampleofsize2iscomplete.2.Usingthelast3-digitsofeach5-digitgroupingprovidestherandomnumbers:601,022,448,147,229,553,147,289,209Numbersgreaterthan350donotapplyandthe147canonlybeusedonce.Thus,thesimplerandomsampleoffourincludes22,147,229,and289.3.459,147,385,113,340,401,215,2,33,3484.a.6,8,5,4,1IBM,Microsoft,Intel,GeneralElectric,AT&TN!10!3,628,500b.252nNn!()!5!(105)!(120)(120)5.283,610,39,254,568,353,602,421,638,1646.2782,493,825,1807,2897.108,290,201,292,322,9,244,249,226,125,(continuingatthetopofcolumn9)147,and113.8.13,8,23,25,18,5Thesecondoccurrencesofrandomnumbers13and25areignored.Washington,Clemson,Oklahoma,Colorado,USCandWisconsin9.511,791,99,671,152,584,45,783,301,568,754,75010.finite,infinite,infinite,infinite,finite5411.a.xxn/9i62()xxib.sn12222222()xx=(-4)+(-1)+1(-2)+1+5=48i48s=31.6113-129 12.a.p=75/150=.50b.p=55/150=.366746513.a.xxn/93i5b.x()xx()xx2iii94+11100+74985-86494+1192-11Totals46501162()xx116is539.n1414.a.149/784=.19b.251/784=.32c.Totalreceivingcash=149+219+251=619619/784=.7970015.a.xxn/7yearsi102()xx20.2ib.s1.5yearsn110116.p=1117/1400=.8017.a.595/1008=.59b.332/1008=.33c.81/1008=.0818.a.Ex()200b.//n501005xc.NormalwithE(x)=200and=5xd.Itshowstheprobabilitydistributionofallpossiblesamplemeansthatcanbeobservedwithrandomsamplesofsize100.Thisdistributioncanbeusedtocomputetheprobabilitythatxiswithinaspecifiedfrom13-130 19.a.ThesamplingdistributionisnormalwithE(x)==200//n501005xFor5,(x-)=5x5z15xArea=.3413x2.6826b.For10,(x-)=10x10z25xArea=.4772x2.954420./nx25/.50354x25/.100250x25/.150204x25/.200177xThestandarderrorofthemeandecreasesasthesamplesizeincreases.21.a.//.n1050141xb.n/N=50/50,000=.001Use//.n1050141xc.n/N=50/5000=.01Use//.n1050141xd.n/N=50/500=.10Nn5005010Use134.UsexN1n500150Note:Onlycase(d)wheren/N=.10requirestheuseofthefinitepopulationcorrectionfactor.Notethatisapproximatelythesameeventhoughthepopulationsizevariesfrominfiniteto500.x13-131 22.a.Usingthecentrallimittheorem,wecanapproximatethesamplingdistributionofxwithanormalprobabilitydistributionprovidedn30.b.n=30//.n5030913xx400n=40//.n5040791x40023.a.//.n1650226xFor2,()x2x2z088.226.xArea=.3106xx2.621216b.160.x100x2z125.160.xArea=.3944x2.788816c.113.x200x2z177.113.xArea=.4616x2.923213-132 16d.080.x400x2z250.080.xArea=.4938x2.9876e.Thelargersampleprovidesahigherprobabilitythatthesamplemeanwillbewithin2of.24.a.//.n40006051640xx51,800E(x)ThenormaldistributionisbasedontheCentralLimitTheorem.b.Forn=120,E(x)remains$51,800andthesamplingdistributionofxcanstillbeapproximatedbyanormaldistribution.However,isreducedto4000/120=365.15.xc.Asthesamplesizeisincreased,thestandarderrorofthemean,,isreduced.Thisappearslogicalxfromthepointofviewthatlargersamplesshouldtendtoprovidesamplemeansthatareclosertothepopulationmean.Thus,thevariabilityinthesamplemean,measuredintermsof,shouldxdecreaseasthesamplesizeisincreased.//.n40006051640x51,30051,80052,30025.a.13-133x 52,300-51,800z==+.97516.40Area=.3340x2.6680b.//.n400012036515x52,300-51,800z==+1.37365.15Area.4147x2.829426.a.AnormaldistributionEx().120/.n010/.500014x122120..b.z141.Area=0.42070014.118120..z141.Area=0.42070014.probability=0.4207+0.4207=0.8414121120..c.z071.Area=0.26120014.119120..z071.Area=0.26120014.probability=0.2612+0.2612=0.522427.a.E(x)=1017//.n100751155x10271017b.z0.87Area=0.307811.5510071017z0.87Area=0.307811.55probability=0.3078+0.3078=0.615610371017c.z1.73Area=0.458211.559971017z1.73Area=0.458211.5513-134 probability=0.4582+0.4582=0.9164x34,00028.a.z/nError=x-34,000=250250n=30z==.68.2518x2=.50362000/30250n=50z==.88.3106x2=.62122000/50250n=100z==1.25.3944x2=.78882000/100250n=200z==1.77.4616x2=.92322000/200250n=400z==2.50.4938x2=.98762000/400b.Alargersampleincreasestheprobabilitythatthesamplemeanwillbewithinaspecifieddistancefromthepopulationmean.Inthesalaryexample,theprobabilityofbeingwithin250ofrangesfrom.5036forasampleofsize30to.9876forasampleofsize400.29.a.E(x)=982/n210/4033.2xx100z3.01/n210/40.4987x2=.9974x25b.z.75/n210/40.2734x2=.5468c.Thesamplewithn=40hasaveryhighprobability(.9974)ofprovidingasamplemeanwithin$100.However,thesamplewithn=40onlyhasa.5468ofprovidingasamplemeanwithin$25.Alargersamplesizeisdesirableifthe$25isneeded.30.a.Normaldistribution,E(x)=166,500/,n42000/1004200xx10000,b.z238.(.4913x2)=.9826/n4200,13-135 c.$5000z=5000/4200=1.19(.3830x2)=.7660$2500z=2500/4200=.60(.2257x2)=.4514$1000z=1000/4200=.24(.0948x2)=.1896d.Increasesamplesizetoimproveprecisionoftheestimate.Samplesizeof100onlyhasa.4514probabilityofbeingwithin$2,500.31.a.//.n52003094939x1000z105.Area=0.353194939.Probability=0.3531x2=0.7062b.//.n52005073539x1000z136.Area=0.413173539.Probability=0.4131x2=0.8262c.//n5200100520x1000z192.Area=0.4726520Probability=0.4726x2=0.9452d.Recommendn=10032.a.n/N=40/4000=.01<.05;therefore,thefinitepopulationcorrectionfactorisnotnecessary.b.WiththefinitepopulationcorrectionfactorNn40004082.129.xN1n4000140Withoutthefinitepopulationcorrectionfactor/.n130xIncludingthefinitepopulationcorrectionfactorprovidesonlyaslightlydifferentvalueforthanxwhenthecorrectionfactorisnotused.c.x2z154.130..130Area.438213-136 x2.876433.a.E(p)=p=.40pp().(10400.)60b.00490.pn100c.NormaldistributionwithE(p)=.40and=.0490pd.Itshowstheprobabilitydistributionforthesampleproportionp.34.a.E(p)=.40pp().(10400.)6000346.pn200pp003.z087.00346.pArea.3078x2.6156b.pp005.z145.00346.pArea.4265x2.8530pp()135.pn(.)(.)05504500497.p100(.)(.)05504500352.p200(.)(.)05504500222.p500(.)(.)05504500157.p1000decreasesasnincreasesp(.)(.)03007036.a.00458.p10013-137 pp004.z087.00458.pArea=0.3078x2=0.6156(.)(.)030070b.00324.p200pp004.z123.00324.pArea=0.3907x2=0.7814(.)(.)030070c.00205.p500pp004.z195.00205.pArea=0.4744x2=0.9488(.)(.)030070d.00145.p1000pp004.z276.00145.pArea=0.4971x2=0.9942e.Withalargersample,thereisahigherprobabilitypwillbewithin.04ofthepopulationproportionp.37.a.pp().(10300.)7000458.pn100p0.3013-138 Thenormaldistributionisappropriatebecausenp=100(.30)=30andn(1-p)=100(.70)=70arebothgreaterthan5.b.P(.20p.40)=?.40-.30z==2.18.0458Area.4854x2.9708c.P(.25p.35)=?.35-.30z==1.09.0458Area.3621x2.724238.a.E(p)=.76pp().(.)1076107600214.pn400079076..b.z140.Area=0.419200214.073076..z140.Area=0.419200214.probability=0.4192+0.4192=0.8384pp().(.)10761076c.00156.pn750079076..z192.Area=0.472600156.073076..z192.Area=0.472600156.probability=0.4726+0.4726=0.945239.a.NormaldistributionE(p)=.50pp(1)(.50)(1.50).0206pn58913-139 pp.04b.z1.94.0206p.4738x2=.9476pp.03c.z1.46.0206p.4279x2=.8558pp.02d.z.97.0206p.3340x2=.668040.a.NormaldistributionE(p)=0.25pp()(10.25)(0.75)00306.pn200003.b.z098.Area=0.336500306.probability=0.3365x2=0.6730005.c.z163.Area=0.448400306.probability=0.4484x2=0.896841.a.E(p)=0.37pp()(10371037.)(.)00153.pn1000040037..b.z196.Area=0.475000153.034037..z196.Area=0.475000153.probability=0.4750+0.4750=0.9500pp()(10371037.)(.)c.00216.pn50013-140 040037..z139.Area=0.417700216.034037..z139.Area=0.417700216.probability=0.4177+0.4177=0.835442.a.pp().1015(0.85)00505.pn50p0.15b.P(.12p.18)=?.18-.15z==.59.0505Area.2224x2.4448c.P(p.10)=?.10-.15z==-.99.0505Area.3389+.5000.838943.a.E(p)=0.17pp()(10171017.)(.)001328.pn800019017..b.z151.Area=0.4345001328.034037..z151.Area=0.4345001328.13-141 probability=0.4345+0.4345=0.8690pp()(10171017.)(.)c.00094.pn1600019017..z213.Area=0.483400094.015017..z213.Area=0.483400094.probability=0.4834+0.4834=0.966844.112,145,73,324,293,875,318,61845.a.NormaldistributionE(x)=31.2.17xn50x.25b.z1.47/1n.2/50.4292x2=.858446.a.NormaldistributionE(x)=31.512170.xn501b.z059.Area=0.2224170.probability=0.2224x2=0.44483c.z177.Area=0.4616170.probability=0.4616x2=0.923247.a.E(x)=$24.07480.044.xn12013-142 050.z114.Area=0.3729044.probability=0.3729x2=0.7458100.b.z228.Area=0.4887044.probability=0.4887x2=0.97746048.a.849.xn50b.z=(115-115)/8.49=0Area=.5000c.z=5/8.49=.59Area=.2224z=-5/8.49=-.59Area=.2224probability=.2224+.2224=.444860d.6xn100z=5/6=.83Area=.2967z=-5/6=-.83Area=.2967probability=.2967+.2967=.5934Nn49.a.xN1nN=20002000501442011.x2000150N=50005000501442026.x5000150N=10,0001000050,1442031.x100001,50Note:Withn/N.05forallthreecases,commonstatisticalpracticewouldbetoignore144thefinitepopulationcorrectionfactoranduse2036.foreachcase.x50b.N=200013-143 25z==1.2420.11Area.3925x2.7850N=500025z123.2026.Area.3907x2.7814N=10,00025z==1.2320.31Area.3907x2.7814Allprobabilitiesareapproximately.7850050.a.20xnn2n500/20=25andn=(25)=625b.For25,25z==1.2520Area.3944x2.788851.Samplingdistributionofxxn300.050.05x1.92.113-144 1.9+2.1==22Theareabetween=2and2.1mustbe.45.Anareaof.45inthestandardnormaltableshowsz=1.645.Thus,2120..1645./30Solvefor(.)0130033.1645.52.a.E(p)=0.74pp()(10741074.)(.)0031.pn200b.z=.04/.031=1.29Area=.4015z=-.04/.031=-1.29Area=.4015probability=.4015+.4015=.8030c.z=.02/.031=.64Area=.2389z=-.02/.031=-.64Area=.2389probability=.2389+.2389=.4778pp()(10.)40(0.)6053.00245.pn400P(p.375)=?.375-.40z==-1.02.0245Area.3461P(p.375)=.3461+.5000=.8461pp(1)(.71)(1.71)54.a..0243pn350pp.05z2.06.0243p.4803x2=.960613-145 pp.75.71b.z1.65.0243pArea=.4505P(p.75)=.5000-.4505=.049555.a.NormaldistributionwithE(p)=.15andpp()(10.15)(0.85)00292.pn150b.P(.12p.18)=?.18-.15z==1.03.0292Area.3485x2.6970pp(12).(.)57556.a..0625pnnSolveforn.(.)2575n482(.0625)b.NormaldistributionwithE(p)=.25and=.0625xc.P(p.30)=?.30.25z.80.0625Area.2881ThusP(.25p.30)=.2881andP(p.30)=.5000-.2881=.2119Chapter8IntervalEstimationLearningObjectives13-146 1.Knowhowtoconstructandinterpretanintervalestimateofapopulationmeanand/orapopulationproportion.2.Understandtheconceptofasamplingerror.3.Beabletouseknowledgeofasamplingdistributiontomakeprobabilitystatementsaboutthesamplingerror.4.Understandandbeabletocomputethemarginoferror.5.Learnaboutthetdistributionanditsuseinconstructinganintervalestimateforapopulationmean.6.Beabletodeterminethesizeofasimplerandomsamplenecessarytoestimateapopulationmeanand/orapopulationproportionwithaspecifiedlevelofprecision.7.Knowthedefinitionofthefollowingterms:confidenceintervalprecisionconfidencecoefficientsamplingerrorconfidencelevelmarginoferrordegreesoffreedomSolutions:1.a.//.n540079xb.At95%,zn/.1965(/)40155.2.a.321.645(/650)13-147 321.4(30.6to33.4)b.321.96(/650)321.66(30.34to33.66)c.322.576(/650)322.19(29.81to34.19)3.a.801.96(/)1560803.8(76.2to83.8)b.801.96(/15120)802.68(77.32to82.68)c.Largersampleprovidesasmallermarginoferror.4.1261.96(/)sn16.071.964n1.96(16.07)n7.8744n625.a.1.96/n1.96(5.00/49)1.40b.24.801.40or(23.40to26.20)6.x1.96(/)sn3691.96(/50250)3696.20(362.80to375.20)7.xz(/)n.0253.371.96(.28/120)3.37.05(3.32to3.42)13-148 8.a.xz/2n12,0001.645(2,200/245)12,000231(11,769to12,231)b.12,0001.96(2,200/245)12,000275(11,725to12,275)c.12,0002.576(2,200/245)12,000362(11,638to12,362)d.Intervalwidthmustincreasesincewewanttomakeastatementaboutwithgreaterconfidence.9.a.Usingacomputer,x=$12.41b.Usingacomputer,s=3.64c.x1.96(/)sn12.411.96(./36460)12.410.92(11.49to13.33)s10.xz.025n3.457.751.961807.75.50(7.25to8.25)11.UsingMinitabweobtainedasamplestandarddeviationof2.163.Theconfidenceintervaloutputisshownbelow:THEASSUMEDSIGMA=2.16NMEANSTDEVSEMEAN95.0PERCENTC.I.Miami506.3402.1630.306(5.740,6.940)The95%confidenceintervalestimateis5.74to6.94.x114i12.a.x3.8minutesn302()xxib.s2.26minutesn113-149 s2.26MarginofError=z1.96.81minutes.025n30sc.xz.025n3.8.81(2.99to4.61)13.a..95b..90c..01d..05e..95f..8514.a.1.734b.-1.321c.3.365d.-1.761and+1.761e.-2.048and+2.0488015.a.xxn/10i82()xx84ib.s3464.n181c.With7degreesoffreedom,t.025=2.365xt.025(/)sn102.365(.3464/8)102.90(7.10to12.90)16.a.17.251.729(./3320)17.251.28(15.97to18.53)b.17.252.09(./3320)13-150 17.251.54(15.71to18.79)c.17.252.861(./3320)17.252.11(15.14to19.36)17.At90%,80t.05(/)snwithdf=17t.05=1.740801.740(/)1018804.10(75.90to84.10)At95%,802.11(/)1018withdf=17t.05=2.110804.97(75.03to84.97)x18.96i18.a.x$1.58n122()xx.239ib.s.1474n1121c.t.025=2.201xt.025(/)sn1.582.201(.1474/12)1.58.09(1.49to1.67)19.xxn/.653minutesi2()xxis054minutes.n1xt.025(/)sn6.532.093(./05420)6.53.25(6.28to6.78)20.a.22.41.96(/561)22.41.25(21.15to23.65)b.Withdf=60,t.025=2.00013-151 22.42(/561)22.41.28(21.12to23.68)c.xt.025(/)snConfidenceintervalsareessentiallythesameregardlessofwhetherzortisused.x864i21.x$108n82()xx654is9.6658n181t.025=2.365xt.025(/)sn1082.365(9.6658/8)1088.08(99.92to116.08)22.a.Usingacomputer,x=6.86s=0.78b.xt.025(/)snt.025=2.064df=246.862.064(./07825)6.860.32(6.54to7.18)2222z(.)()19625.02523.n9604.Usen9722E524.a.Planningvalueof=Range/4=36/4=92222z(.)()1969.025b.n3457.Usen3522E322(.)()1969c.nn7779.Use782222(.)(.)19668225.nn7941.Use802(.)1522(.1645)(.)682nn3147.Use322213-152 2222z(1.96)(9400)26.a.n339.44Use34022E(1000)22(1.96)(9400)b.n1357.78Use13582(500)22(1.96)(9400)c.n8486.09Use848720022(.)(,1962000)27.a.nn6147.Use622()50022(.)(,1962000)b.nn38416.Use3852()20022(.)(,1962000)c.nn153664.Use15372()1002222z(1.645)(220)28.a.n52.39Use5322E(50)22(1.96)(220)b.n74.37Use752(50)22(2.576)(220)c.n128.47Use1292(50)d.Mustincreasesamplesizetoincreaseconfidence.22(.)(.)19662529.a.nn3752.Use382222(.)(.)196625b.nn15006.Use1512122(.)(.)1967830.nn5843.Use592231.a.p=100/400=0.25pp(10).(.)25075b.00217.n40013-153 pp()1c.pz.025n.251.96(.0217).25.0424(.2076to.2924)070030.(.)32.a..701.645800.70.0267(.6733to.7267)070030.(.)b..701.96800.70.0318(.6682to.7318)22zpp()1(.)(.)(.)196035065.02533.n34959.Usen35022E(.)00534.Useplanningvaluep=.502(.)(.)(.)196050050nn106711.Use10682(.)00335.a.p=562/814=0.6904pp()106904106904.(.)b.1645.1645.00267.n814c.0.69040.0267(0.6637to0.7171)36.a.p=152/346=.4393pp(1).4393(1.4393)b..0267pn346pz.025p.43931.96(.0267).4393.0523(.3870to.4916)pp()137.p196.n182p.2865013-154 (.)(.)028072.281.966500.280.0345(0.2455to0.3145)pp()1(.)(.)02607438.a.196.196.00430.n400b.0.260.0430(0.2170to0.3030)2196026074.(.)(.)c.nn82125.Use8222(.)00322zpp(1)(1.96)(.33)(1.33).02539.a.n943.75Use94422E(.03)22zpp(1)(2.576)(.33)(1.33).005b.n1630.19Use163122E(.03)40.a.p=255/1018=0.2505(.02505102505)(.)b.1.96=0.02661018pp(1).16(1.16)41..0102pn1285MarginofError=1.96=1.96(.0102)=.02p.161.96p.16.02(.14to.18)pp(1).50(1.50)42.a..0226pn491z=1.96(.0226)=.0442.025p2zpp(1).025b.n2E21.96(.50)(1.50)Septembern600.25Use6012.0421.96(.50)(1.50)Octobern1067.11Use10682.0313-155 21.96(.50)(1.50)Novembern24012.0221.96(.50)(1.50)Pre-Electionn96042.01219605105.(.)(.)43.a.nn60025.Use6012(.)004b.p=445/601=0.7404(.0740402596)(.)c.0.74041.966010.74040.0350(0.7054to0.7755)s20,50044.a.z1.962009.025n400b.xz.025(/)sn50,0002009(47,991to52,009)45.a.xz.025(/)sn252.451.96(./745064)252.4518.25or$234.20to$270.70b.Yes.thelowerlimitforthepopulationmeanatNiagaraFallsis$234.20whichisgreaterthan$215.60.46.a.Usingacomputer,x=49.8minutesb.Usingacomputer,s=15.99minutesc.x1.96(/)sn49.81.96(./1599200)49.82.22(47.58to52.02)47.a.Usingacomputer,x=16.8ands=4.25With19degreesoffreedom,t.025=2.093x2.093(/)sn13-156 16.82.093(./42520)16.81.99(14.81to18.79)b.Usingacomputer,x=24.1ands=6.2124.12.093(./62120)24.12.90(21.2to27.0)c.16.8/24.1=0.697or69.7%orapproximately70%13248.a.xxn/.132i102()xx5476.ib.s78.n19c.Withdf=9,t.025=2.262xt.025(/)sn13.22.262(./7810)13.25.58(7.62to18.78)d.The5.58showspoorprecision.Alargersamplesizeisdesired.2196045.(.)49.nn7779.Use7821022(.)(.)2332650.nn367.Use372122(.)()196851.nn6147.Use622222(.)()25768nn10617.Use1072222(.)(196675)52.nn17503.Use1762100pp()153.a.p196.n13-157 (.)(.)0470530.471.964500.470.0461(0.4239to0.5161)(.)(.)047053b.0.472.5764500.470.0606(0.4094to0.5306)c.Themarginoferrorbecomeslarger.54.a.p=200/369=0.5420pp()1(.0542004580)(.)b.196.196.00508.n369c.0.54200.0508(0.4912to0.5928)55.a.p=504/1400=.36(.)(.)036064b.196.00251.14002(.)(.)(.)23307003056.a.nn126674.Use12672(.)0032(.)(.)(.)233050050b.nn150803.Use15092(.)00357.a.p=110/200=0.55(.)(.)0550450.551.96200.55.0689(.4811to.6189)2(.)(.)(.)196055045b.nn38032.Use3812(.)00558.a.p=340/500=.68pp(1).68(1.68)b..0209pn500pz.025p.681.96(.0209)13-158 .68.0409(.6391to.7209)2(.)(.)(.)196030759.a.nn201684.Use20172(.)002b.p=520/2017=0.2578pp()1c.p196.n(.0257807422)(.)0.25781.9620170.25780.0191(0.2387to0.2769)60.a.p=618/1993=.3101pp()1b.p196.1993(.0310106899)(.)0.31011.961993.3101.0203(.2898to.3304)2zp()1pc.n2E2(.)(.1960310106899)(.)zn821864.Use82192(.)001No;thesampleappearsunnecessarilylarge.The.02marginoferrorreportedinpart(b)shouldprovideadequateprecision.Chapter9HypothesisTestingLearningObjectives1.Learnhowtoformulateandtesthypothesesaboutapopulationmeanand/orapopulationproportion.2.Understandthetypesoferrorspossiblewhenconductingahypothesistest.13-159 3.Beabletodeterminetheprobabilityofmakingvariouserrorsinhypothesistests.4.Knowhowtocomputeandinterpretp-values.5.Beabletodeterminethesizeofasimplerandomsamplenecessarytokeeptheprobabilityofhypothesistestingerrorswithinacceptablelimits.6.Knowthedefinitionofthefollowingterms:nullhypothesisone-tailedtestalternativehypothesistwo-tailedtesttypeIerrorp-valuetypeIIerroroperatingcharacteristiccurvecriticalvaluepowercurvelevelofsignificanceSolutions:1.a.H0:600Manager’sclaim.Ha:>600b.Wearenotabletoconcludethatthemanager’sclaimiswrong.c.Themanager’sclaimcanberejected.Wecanconcludethat>400.2.a.H0:14Ha:>14Researchhypothesis13-160 b.Thereisnostatisticalevidencethatthenewbonusplanincreasessalesvolume.c.Theresearchhypothesisthat>14issupported.Wecanconcludethatthenewbonusplanincreasesthemeansalesvolume.3.a.H0:=32SpecifiedfillingweightHa:32Overfillingorunderfillingexistsb.Thereisnoevidencethattheproductionlineisnotoperatingproperly.Allowtheproductionprocesstocontinue.c.Conclude32andthatoverfillingorunderfillingexists.Shutdownandadjusttheproductionline.4.a.H0:220Ha:<220Researchhypothesistoseeifmeancostislessthan$220.b.Weareunabletoconcludethatthenewmethodreducescosts.c.Conclude<220.Considerimplementingthenewmethodbasedontheconclusionthatitlowersthemeancostperhour.5.a.TheTypeIerrorisrejectingH0whenitistrue.Inthiscase,thiserroroccursiftheresearcherconcludesthatthemeannewspaper-readingtimeforindividualsinmanagementpositionsisgreaterthanthenationalaverageof8.6minuteswheninfactitisnot.b.TheTypeIIerrorisacceptingH0whenitisfalse.Inthiscase,thiserroroccursiftheresearcherconcludesthatthemeannewspaper-readingtimeforindividualsinmanagementpositionsislessthanorequaltothenationalaverageof8.6minuteswheninfactitisgreaterthan8.6minutes.6.a.H0:1Thelabelclaimorassumption.Ha:>1b.Claiming>1whenitisnot.Thisistheerrorofrejectingtheproduct’sclaimwhentheclaimistrue.c.Concluding1whenitisnot.Inthiscase,wemissthefactthattheproductisnotmeetingitslabelspecification.7.a.H0:8000Ha:>8000Researchhypothesistoseeiftheplanincreasesaveragesales.b.Claiming>8000whentheplandoesnotincreasesales.Amistakecouldbeimplementingtheplanwhenitdoesnothelp.c.Concluding8000whentheplanreallywouldincreasesales.Thiscouldleadtonotimplementingaplanthatwouldincreasesales.8.a.H0:220Ha:<22013-161 b.Claiming<220whenthenewmethoddoesnotlowercosts.Amistakecouldbeimplementingthemethodwhenitdoesnothelp.c.Concluding220whenthemethodreallywouldlowercosts.Thiscouldleadtonotimplementingamethodthatwouldlowercosts.9.a.z=-1.645RejectH0ifz<-1.645x9.4610b.z1.91sn/2/50RejectH0;concludeHaistrue.10.a.z=2.05RejectH0ifz>2.05x16.515b.z1.36sn/7/40c.Areafromz=0toz=1.36is.4131p-value=.5000-.4131=.0869d.DonotrejectH011.RejectH0ifz<-1.645x2225a.z2.50RejectH0/n12/1002425b.z.83DonotrejectH012/10023.525c.z1.25DonotrejectH012/10022.825d.z1.83RejectH012/10012.a.p-value=.5000-.4656=.0344RejectH0b.p-value=.5000-.1736=.3264DonotrejectH0c.p-value=.5000-.4332=.0668DonotrejectH0d.z=3.09isthelargesttablevaluewith.5000-.4990=.001areaintail.Forz=3.30,thep-valueislessthan.001orapproximately0.RejectH0.13-162 e.Sincezistotheleftofthemeanandtherejectionregionisintheuppertail,p-value=.5000+.3413=.8413.DonotrejectH0.13.a.H0:1056Ha:<1056b.RejectH0ifz<-1.645x91010560c.z1.83sn/1600/400d.RejectH0andconcludethatthemeanrefundof“lastminute”filersislessthan$1056.e.p-value=.5000-.4664=.033614.a.z.01=2.33RejectH0ifz>2.33x7.256.70b.z3.11sn/2.5/200c.RejectH0;concludethemeantelevisionviewingtimeperdayisgreaterthan6.70.15.a.z.05=1.645RejectH0ifz<-1.645x930010,192b.z1.98sn/4500/100c.RejectH0;concludethatthemeansalespriceofusedcarsislessthanthenationalaverage.16.a.H0:Ha:<13b.z.01=2.33RejectH0ifz<-2.33x10.813c.z2.88sn/9.2/145d.RejectH0;concludeCanadianmeaninternetusageislessthan13hourspermonth.Note:p-value=.00217.a.H0:15Ha:>1513-163 x1715b.z2.96sn/4/35c.p-value=.5000-.4985=.0015d.RejectH0;thepremiumrateshouldbecharged.18.a.H0:5.72Ha:>5.72x5.985.72b.z2.12sn/1.24/102c.p-value=.5000-.4830=.0170d.p-value<;rejectH0.ConcludeteensinChicagohaveameanexpendituregreaterthan5.72.19.a.H0:181,900Ha:<181,900x166,400181,900b.z2.93sn/33,500/40c.p-value=.5000-.4983=.0017d.p-value<;rejectH0.ConcludemeansellingpriceinSouthislessthanthenationalmeansellingprice.20.a.H0:37,000Ha:>37,000x38,10037,000b.z1.47sn/5200/48c.p-value=.5000-.4292=.0708d.p-value>;donotrejectH0.CannotconcludepopulationmeansalaryhasincreasedinJune2001.21.a.RejectH0ifz<-1.96orz>1.96x10810.b.z240.sn/25./36RejectH0;concludeHaistrue.22.a.RejectH0ifz<-2.33orz>2.3313-164 x14215.b.z113.sn/55/0c.p-value=(2)(.5000-.3708)=.2584d.DonotrejectH023.RejectH0ifz<-1.96orz>1.962225a.z2.68RejectH010/802725b.z1.79DonotrejectH010/8023.525c.z1.34DonotrejectH010/802825d.z2.68RejectH010/8024.a.p-value=2(.5000-.4641)=.0718DonotrejectH0b.p-value=2(.5000-.1736)=.6528DonotrejectH0c.p-value=2(.5000-.4798)=.0404RejectH0d.approximately0RejectH0e.p-value=2(.5000-.3413)=.3174DonotrejectH025.a.z.025=1.96RejectH0ifz<-1.96orz>1.96x38.539.2b.z1.54sn/4.8/112c.DonotrejectH0.Cannotconcludeachangeinthepopulationmeanhasoccurred.d.p-value=2(.5000-.4382)=.123626.a.H0:=8Ha:8RejectH0ifz<-1.96orifz>1.96x758.0b.z171.sn/32./120c.DonotrejectH0;cannotconcludethemeanwaitingtimediffersfromeightminutes.27.a.H0:=16Continueproduction13-165 Ha:16ShutdownRejectH0ifz<-1.96orifz>1.96x16.32160b.z2.19/.n8/30RejectH0andshutdownforadjustment.x15.82160c.z1.23/.n8/30DonotrejectH0;continuetorun.d.Forx=16.32,p-value=2(.5000-.4857)=.0286Forx=15.82,p-value=2(.5000-.3907)=.218628.a.H0:=1075Ha:1075x11601075b.z1.43sn/840/200c.p-value=2(.5000-.4236)=.1528d.DonotrejectH0.Cannotconcludeachangeinmeanamountofcharitablegiving.29.a.H0:=15.20Ha:15.20x14301520..0z106.sn/53/5b.p-value=2(.5000-.3554)=.2892c.DonotrejectH0;thesampledoesnotprovideevidencetoconcludethattherehasbeenachange.30.a.H0:=26,133Ha:26,133x25,45726,133b.z2.09sn/7600/410c.p-value=2(.5000-.4817)=.0366d.p-value<;rejectH0.ConcludepopulationmeanwageinCollierCountydiffersfromthestatemeanwage.13-166 31.a.xz.025n1809351.9620093525or910to960Since900isnotintheinterval,rejectH0andconclude900.b.RejectH0ifz<-1.96orifz>1.96x9359000z2.75/n180/200RejectH0c.p-value=2(.5000-.4970)=.006032.a.Theupper95%confidencelimitiscomputedasfollows:sxz.05n.6014.501.64514.6636Thus,weare95%confidentthatis$14.66perhourorless.b.Since$15.00isnotintheinterval$14.66perhourorless,werejectH0.Concludethatthemeanwagerateislessthan$15.00.33.a.With15degreesoffreedom,t.05=1.753RejectH0ift>1.753x11100b.t133.DonotrejectH0sn//31634.a.xx/n=108/6=18i()xx10ib.s1.414n161c.RejectH0ift<-2.571ort>2.571x18200d.t346.sn/.1414/6e.RejectH0;concludeHaistrue.13-167 35.RejectH0ift<-1.7211315a.t117.DonotrejectH082/211515.b.t205.RejectH082/21515c.t0DonotrejectH082/21915d.t235.DonotrejectH082/236.Usethetdistributionwith15degreesoffreedoma.p-value=.01RejectH0b.p-value=.10DonotrejectH0c.p-valueisbetween.025and.05RejectH0d.p-valueisgreaterthan.10DonotrejectH0e.p-valueisapproximately0RejectH037.a.H0:3.00Ha:3.00b.t.025=2.262RejectH0ift<-2.262orift>2.262x28ic.x2.80n102()xx.44id.s.70n1101x2.803.00e.t.90sn/.70/10f.DonotrejectH0;cannotconcludethepopulationmeansearningpersharehaschanged.g.t.10=1.383p-valueisgreaterthan.10x2=.2013-168 Actualp-value=.391638.a.t.025=2.06424degreesoffreedomRejectH0ift<-2.064orift>2.064x84.5090b.t1.90sn/14.50/25c.DonotrejectH0;cannotconcludethemeanexpenditureinCorningdiffersfromtheU.S.meanexpenditure.39.a.t.05=1.8957degreesoffreedomx475ib.x59.375n82()xx123.87ic.s4.21n181x59.3855d.t2.94sn/4.21/8c.RejectH0;concludethatthemeannumberofhoursworkedexceeds55.40.a.H0:4000Ha:4000b.t.05=2.16013degreesoffreedomRejectH0ift<-2.160orift>2.160x41204000c.t1.63sn/275/14d.DonotrejectH0;CannotconcludethatthemeancostinNewCitydiffersfrom$4000.e.With13degreesoffreedomt.05=1.771t.10=1.3501.63isbetween1.350and1.771.Thereforethep-valueisbetween.10and.20.41.a.H0:280Ha:>28013-169 b.286.9-280=6.9yardsc.t.05=1.860with8degreesoffreedomx286.9280d.t2.07sn/10/9e.RejectH0;ThepopulationmeandistanceofthenewdriverisgreaterthantheUSGAapproveddriver..f.t.05=1.860t.025=2.306p-valueisbetween.025and.05Actualp-value=.036142.a.H0:2Ha:>2b.With9degreesoffreedom,rejectH0ift>1.833c.xx/n=24/10=2.4i2()xx2.40id.s.516n19x2.420e.t2.45sn/.516/10f.RejectH0andclaimisgreaterthan2hours.Forcostestimatingpurposes,considerusingmorethan2hoursoflabortime.g.t.025=2.262,t.01=2.821p-valueisbetween.025and.01.43.a.RejectH0ifz>1.645.(.)5050b..0354p200pp..5750z198Reject.H0.0354p44.a.RejectH0ifz<-1.96orz>1.9613-170 .(.)2080b..02p400pp..17520z125..02pc.p-value=2(.5000-.3944)=.2122d.DonotrejectH0.45.RejectH0ifz<-1.645.(.)7525a..0250p300pp..6875z280..025pp-value=.5000-.4974=.0026RejectH0.pp..7275b.z120..025pp-value=.5000-.3849=.1151DonotrejectH0.pp..7075c.z200..025pp-value=.5000-.4772=.0228RejectH0.pp..7775d.z.80.025pp-value=.5000+.2881=.7881DonotrejectH0.46.a.H0:p.40Ha:p>.40b.RejectH0ifz>1.645c.p=188/420=.447613-171 pp(1).40(1.40).0239pn420pp.4476.40z1.99.0239pd.RejectH0.Concludethattherehasbeenanincreaseintheproportionofusersreceivingmorethantene-mailsperday.47.a.z.05=1.645RejectH0ifz<-1.645b.p=52/100=.52pp(1).64(1.64).0480pn100pp.52.64z2.50.0480pc.RejectH0.Concludelessthan64%ofshoppersbelievesupermarketketchupisasgoodasthenationalbrand.d.p-value=.5000-.4938=.006248.a.p=285/500=.57pp(1).50(1.50)b..0224pn500pp.57.50z3.13.0224pc.z=3.13isnotinthetable.Closestvalueisz=3.09.Thus,p-valueisapproximately.5000-.4990=.001d.p-value<.01,RejectH0.Over50%preferBurgerKingfries.e.Yes;thestatisticalevidenceshowsBurgerKingfriesarepreferred.Thegive-awaywasagoodwaytogetpotentialcustomerstotrythenewfries.49.a.H0:p=.48Ha:p.48b.z.025=1.96RejectH0ifz<-1.96orifz>1.9613-172 c.p=368/800=.45pp(1).48(1.48)d..0177pn800pp.45.48z1.70.0177pd.DonotrejectH0.Cannotconcludetheproportionofdriverswhodonotstophaschanged.50.a.p=67/105=.6381(about64%)pp(1).50(1.50)b..0488pn105pp.6381.50z2.83.0488pc.p-value=2(.5000-.4977)=.0046d.p-value<.01,rejectH0.Concludepreferenceisforthefourten-hourdayschedule.51.a.H0:p=.44Ha:p.44b.p=205/500=.41pp(1).44(1.44).0222pn500pp.41.44z1.35.0222pp-value=2(.5-.4115)=.1770DonotrejectH0.Cannotconcludethattherehasbeenachangeintheproportionofrepeatcustomers.c.p=245/500=.49pp.49.44z2.25.0222pp-value=2(.5-.4878)=.0244RejectH0.concludethattheproportionofrepeatcustomershaschanged.Thepointestimateofthepercentageofrepeatcustomersisnow49%.13-173 pp(1).75(1.75)52.a..025pn300pp.72.75z1.20.025pb.p-value=.5000-.3849=.1151c.DonotrejectH0.Cannotconcludethemanager"sclaimiswrongbasedonthissampleevidence.53.H0:p.15Ha:p>.15RejectH0ifz>2.33pp(1).15(.85).0160pn500p=88/500=.176pp.176.150z1.63.0160pDonotrejectH0;p.15cannotberejected.Thusthespecialoffershouldnotbeinitiated.p-value=.5000-.4484=.051654.a.H0:p.047Ha:p<.047b.p=35/1182=.0296.047(1.047)c..0062p1182pp.0296.047z2.82.0062pd.p-value=.5000-.4976=.0024e.p-value<,rejectH0.TheerrorrateforBrooksRobinsonislessthantheoverallerrorrate.55.H0:p.20Ha:p<.20RejectH0ifz<-1.64513-174 p=83/596=.1393pp(1).20(1.20).0164pn596pp.1393.20z3.71.0164pp-value0RejectH0;concludethatlessthan20%ofworkerswouldworkforlesspayinordertohavemorepersonalandleisuretime.556..46xn120cHa:<10H0:10.05x10c=10-1.645(5/120)=9.25RejectH0ifx<9.25a.When=9,9.259z.555/120Prob(H0)=(.5000-.2088)=.2912b.TypeIIerrorc.When=8,9.258z2.745/120=(.5000-.4969)=.003113-175 57.RejectH0ifz<-1.96orifz>1.9610.71xn200Ha:20H0:=20Ha:20.025.025xxc120c2c1=20-1.96(10/200)=18.61c2=20+1.96(10/200)=21.39a.=1818.6118z.8610/200=.5000-.3051=.1949b.=22.521.3922.5z1.5710/200=.5000-.4418=.0582c.=2121.3921z.5510/200=.5000+.2088=.708858.a.H0:1513-176 Ha:>15Concluding15whenthisisnottrue.Fowlewouldnotchargethepremiumrateeventhoughtherateshouldbecharged.b.RejectH0ifz>2.33xx150z2.33/4n/35Solveforx=16.58DecisionRule:AcceptH0ifx16.58RejectH0ifx>16.58For=17,16.5817z.624/35=.5000-.2324=.2676c.For=18,16.5818z2.104/35=.5000-.4821=.017959.a.H0:25Ha:<25RejectH0ifz<-2.05xx250z2.05/3n/30Solveforx=23.88DecisionRule:AcceptH0ifx23.88RejectH0ifx<23.8813-177 b.For=23,23.8823z1.613/30=.5000-.4463=.0537c.For=24,23.8824z.223/30=.5000+.0871=.5871d.TheTypeIIerrorcannotbemadeinthiscase.Notethatwhen=25.5,H0istrue.TheTypeIIerrorcanonlybemadewhenH0isfalse.60.a.AcceptingH0andconcludingthemeanaverageagewas28yearswhenitwasnot.b.RejectH0ifz<-1.96orifz>1.96xx280z/n6/100Solvingforx,wefindatz=-1.96,x=26.82atz=+1.96,x=29.18DecisionRule:AcceptH0if26.82x29.18RejectH0ifx<26.82orifx>29.18At=26,26.8226z1.376/100=.5000+.4147=.0853At=27,26.8227z.306/100=.5000+.1179=.617913-178 At=29,29.1829z.306/100=.5000+.1179=.6179At=30,29.1830z1.376/100=.5000-.4147=.0853c.Power=1-at=26,Power=1-.0853=.9147When=26,thereisa.9147probabilitythatthetestwillcorrectlyrejectthenullhypothesisthat=28.61.a.AcceptingH0andlettingtheprocesscontinuetorunwhenactuallyover-fillingorunder-fillingexists.b.DecisionRule:RejectH0ifz<-1.96orifz>1.96indicatesAcceptH0if15.71x16.29RejectH0ifx<15.71orifx>16.29For=16.516.2916.5z1.44.8/30=.5000-.4251=.074913-179 cx16.2916.5c.Power=1-.0749=.9251d.ThepowercurveshowstheprobabilityofrejectingH0forvariouspossiblevaluesof.Inparticular,itshowstheprobabilityofstoppingandadjustingthemachineunderavarietyofunderfillingandoverfillingsituations.Thegeneralshapeofthepowercurveforthiscaseis1.00.75Power.50.25.0015.615.816.016.216.4PossibleValuesofu462.cz152.3316.320.01n5016.3217Atz1.204/50=.5000-.3849=.115116.3218Atz2.974/50=.5000-.4985=.0015IncreasingthesamplesizereducestheprobabilityofmakingaTypeIIerror.13-180 63.a.Accept100whenitisfalse.b.Criticalvaluefortest:75cz1001.645119.510.05n40119.51120Atz.0475/40=.5000-.0160=.4840119.51130c.Atz.8875/40.5000-.3106=.1894d.Criticalvaluefortest:75cz1001.645113.790.05n80113.79120Atz.7475/80=.5000-.2704=.2296113.79130Atz1.9375/80=.5000-.4732=.0268Increasingthesamplesizefrom40to80reducestheprobabilityofmakingaTypeIIerror.2222()zz(1.6451.28)(5)64.n21422()(109)0a2222()zz(1.961.645)(10)65.n32522()(2022)0a66.At0=3,=.01.z.01=2.33Ata=2.9375,=.10.z.10=1.28=.182222()zz(2.331.28)(.18)n108.09Use10922()(32.9375)0a13-181 67.At0=400,=.02.z.02=2.05Ata=385,=.10.z.10=1.28=302222()zz(2.051.28)(30)n44.4Use4522()(400385)0a68.At0=28,=.05.Notehoweverforthistwo-tailedtest,z/2=z.025=1.96Ata=29,=.15.z.15=1.04=62222()zz(1.961.04)(6)/2n32422()(2829)0a69.At0=25,=.02.z.02=2.05Ata=24,=.20.z.20=.84=32222()zz(2.05.84)(3)n75.2Use7622()(2524)0a70.a.H0:45,250Ha:>45,250x47,00045,250b.z2.71sn/6300/95c.p-value=.5000-.4966=.0034d.p-value<;rejectH0.NewYorkCityschoolteachersmusthaveahighermeanannualsalary.71.H0:30Ha:<30RejectH0ifz<–2.33x29.5300z1.96sn/1.8/50p-value=.5000-.4750=.025013-182 DonotrejectH0;thesampleevidencedoesnotsupporttheconclusionthattheFordTaurusprovideslessthan30milespergallon.72.H0:25,000Ha:>25,000RejectH0ifz>1.645x26,00025,0000z2.26sn/2,500/32p-value=.5000-.4881=.0119RejectH0;theclaimshouldberejected.Themeancostisgreaterthan$25,000.73.H0:=120Ha:120Withn=10,useatdistributionwith9degreesoffreedom.RejectH0ift<-2.262oroft>2.262xix118.9n2()xxis4.93n1x118.91200t.71sn/4.93/10DonotrejectH0;theresultsdonotpermitrejectionoftheassumptionthat=120.74.a.H0:=550Ha:550RejectH0ifz<-1.96orifz>1.96x5625500z180.sn//4036DonotrejectH0;theclaimof$550permonthcannotberejected.b.p-value=2(.5000-.4641)=.071875.a.H0:75Ha:>7513-183 RejectH0ifz>1.645x82.5075.000b.z1.58sn/30/40DonotrejectH0;thereisnoevidencetoconcludeanincreaseinmaintenancecostexists.c.p-value=.5000-.4429=.0571Since.0571>.05,donotrejectH0.76.a.H0:72Ha:>72x728072z219.sn//2030p-value=.5000-.4857=.0143b.Sincep-value<.05,rejectH0;themeanidletimeexceeds72minutesperday.77.a.H0:p.60Ha:p>.60RejectH0ifz>1.645pp(1).60(.40).0775pn40p=27/40=.675pp.675.60z.97.0775pDonotrejectH0;thesampleresultsdonotjustifytheconclusionthatp>.60forMidwesterners.b.p-value=.5000-.3340=.166078.a.p=355/546=.6502pp(1).67(1.67)b..0201pn546pp.6502.67z.98.0201pc.p-value=2(.5000-.3365)=.3270d.p-value,donotrejectH0.Theassumptionoftwo-thirdscannotberejected.13-184 79.H0:p.79Ha:p<.79RejectH0ifz<-1.645p=360/500=.72pp..72790z384.p(.7921)(.)500RejectH0;concludethattheproportionislessthan.79in1995.80.a.Theresearchisattemptingtoseeifitcanbeconcludedthatlessthan50%oftheworkingpopulationholdjobsthattheyplannedtohold..(.)5050b..0136p1350..4150z662..0136p-value0RejectH0ifz<-2.33RejectH0;itcanbeconcludedthatlessthan50%oftheworkingpopulationholdjobsthattheyplannedtohold.Themajorityholdjobsduetochance,lackofchoice,orsomeotherunplannedreason..(.)752581..0229p356p=313/356=.88..8875z568..0229p-value0RejectH0;concludep.75.Datasuggestthat88%ofwomenwearshoesthatareatleastonesizetoosmall.82.a.p=330/400=.825pp(1).78(1.78)b..0207pn40013-185 pp.825.78z2.17.0207pc.p-value=2(.5000-.4850)=.03d.p-value<,rejectH0.Arrivalratehaschangedfrom78%.Serviceappearstobeimproving.83.H0:p.90Ha:p<.90RejectH0ifz<-1.645.90(.10).0394p58p=49/58=.845pp.845.90z1.40.0394pp-value=.5000-.4192=.0808DonotrejectH0;thestation’sclaimcannotberejected84.a.p=44/125=.352pp(1).47(1.47)b..0446pn125pp.352.47z2.64.0446pc.p-value=.5000-.4959=.0041d.RejectH0;concludethattheproportionoffoodsamplecontainingpesticideresidueshasbeenreduced.85.a.H0:72Ha:>72RejectH0ifz>1.645xx720z1.645/2n0/30Solveforx=7813-186 DecisionRule:AcceptH0ifx78RejectH0ifx>78For=807880z.5520/30=.5000-.2088=.2912b.For=75,7875z.8220/30=.5000+.2939=.7939c.For=70,H0istrue.InthiscasetheTypeIIerrorcannotbemade.d.Power=1-1.0.8Po.6wer.4.272747678808284PossibleValuesofHoFalse86.H0:15,000Ha:<15,000At0=15,000,=.02.z.02=2.05Ata=14,000,=.05.z.10=1.6452222()zz(2.051.645)(4,000)n218.5Use21922()(15,00014,000)0a87.H0:=12013-187 Ha:120At0=120,=.05.Withatwo-tailedtest,z/2=z.025=1.96Ata=117,=.02.z.02=2.052222()zz(1.962.05)(5)/2n44.7Use4522()(120117)0ab.Examplecalculationfor=118.RejectH0ifz<-1.96orifz>1.96xx1200z/5n/45Solveforx.Atz=-1.96,x=118.54Atz=+1.96,x=121.46DecisionRule:AcceptH0if118.54x121.46RejectH0ifx<118.54orifx>121.46For=118,118.54118z.725/45=.5000+.2642=.2358OtherResults:Ifisz1172.07.0192118.72.2358119-.62.7291121+.62.7291122+.72.2358123-2.07.0192Chapter10StatisticalInferenceaboutMeansandProportionswithTwoPopulations13-188 LearningObjectives1.Beabletodevelopintervalestimatesandconducthypothesistestsaboutthedifferencebetweenthemeansoftwopopulations.2.Knowthepropertiesofthesamplingdistributionofthedifferencebetweentwomeansxx.123.Beabletousethetdistributiontoconductstatisticalinferencesaboutthedifferencebetweenthemeansoftwonormalpopulationswithequalvariances.4.Understandtheconceptanduseofapooledvarianceestimate.5.Learnhowtoanalyzethedifferencebetweenthemeansoftwopopulationswhenthesamplesareindependentandwhenthesamplesarematched.6.Beabletodevelopintervalestimatesandconducthypothesistestsaboutthedifferencebetweentheproportionsoftwopopulations.7.Knowthepropertiesofthesamplingdistributionofthedifferencebetweentwoproportionspp12.Solutions:1.a.xx=13.6-11.6=21213-189 2222ss(.)22()312b.s0595.xx12nn50351221.645(.595)2.98(1.02to2.98)c.21.96(.595)21.17(0.83to3.17)2.a.xx=22.5-20.1=2.41222222()()(nsns111122925.)(72)b.s527.nn21082122F11IF11Ic.ss527..109xx12HGKJHGnn12KJ10816degreesoffreedom,t.025=2.122.42.12(1.09)2.42.31(.09to4.71)3.a.xx//n54691ixx//n42672i2()xx18i1b.s1.901n16112()xx16i2s1.792n1612c.xx=9-7=21222222()()nsns1111225190(.)5179(.)d.s341.nn266212e.With10degreesoffreedom,t.025=2.2282F11IF11Iss341..107xx12HGKJHGnn12KJ6622.228(1.07)22.37(-0.37to4.37)13-190 4.a.xx=1.58-0.98=$0.60122222ss.12.0812b.s.021xx12nn504212x12xz.025sx12x.60±1.96(.021).60±.04(.56to.64)5.a.22.5-18.6=3.9milesperdayb.x12xz/2sx12x2222ss(8.4)(7.4)12s1.58xx12nn50501222.5-18.61.96(1.58)3.93.1or0.6to7.06.LAMiamix6.726.34s2.3742.163xxzs12/2xx122222ss(.)2374(.)216312s0454.xx12nn5050126.72-6.341.96(.454).38.89or-.51to1.277.a.xx=14.9-10.3=4.6years122222ss5.23.812b.s.66xx12nn1008512z.025sxx=1.96(.66)=1.312c.x12xz.025sx12x13-191 4.61.3(3.3to5.9)8.a.xx=15,700-14,500=1,20012b.Pooledvariance2227(700)11(850)s632,0831811s632,083362.88xx12812With18degreesoffreedomt.025=2.10112002.101(362.88)1200762(438to1962)c.Populationsarenormallydistributedwithequalvariances.9.a.n1=10n2=8x=21.2x=22.812s1=2.70s2=3.55xx=21.2-22.8=-1.612Kitchensarelessexpensiveby$1,600.b.x12xz/2sx12xDegreesoffreedom=n1+n2-2=16t.05=1.7462229(2.70)7(3.55)s9.63108211s9.631.47xx12108-1.61.746(1.47)-1.62.57(-4.17to+.97)10.a.x=17.54x=15.3612xx=17.54-15.36=$2.18perhourgreaterforunionworkers.1213-192 22222()()nsns11112214224(.)(19199.)b.s441.nn21520212c.xxts122/xx1211s4.410.72xx12152017.5415.36t(.72)/22.18t(.72)/2Note:Valuesfort.025arenotlistedfor33degreesoffreedom;for30d.f.t.025=2.042andfor40d.f.t.025=2.021.Wewillusethemoreconservativevalueof2.042asanapproximation.2.182.042(.72)2.181.47or0.71to3.652222ss(5.)2()61211.a.s118.xx12nn405012(25.222.8)z2.031.18RejectH0ifz>1.645RejectH0;concludeHaistrueand>0.b.p-value=.5000-.4788=.02122222ss(8.)4(.)761212.a.s131.xx12nn807012()xx()(104106)01212z153.s131.xx12RejectH0ifz<-1.96orz>1.96DonotrejectH0b.p-value=2(.5000-.4370)=.126013.a.xx=1.4–1.0=0.41222222()()(nsns11112274.)(66.)s02523.nn28721213-193 11s0.25230.26xx1287With13degreesoffreedom.t.025=2.16RejectH0ift<-2.16ort>2.16()xx().041212t154.s026.xx12DonotrejectH014.a.H0:µ1-µ2=0Ha:0b.RejectH0ifz<-1.96orifz>1.962222ss16.815.212c.s1.79xx12nn15017512xx12039.335.40z2.18s1.79xx12d.RejectH0;concludethepopulationmeansdiffer.e.p-value=2(.5000-.4854)=.029215.H0:µ1-µ2=0Ha:0RejectH0ifz<-1.96orifz>1.96()xx0(4035)12z2.412222(9)(10)12nn364912RejectH0;customersatthetwostoresdifferintermsofmeanages.p-value=2(.5000-.4920)=.016016.H0:0Ha:>0RejectH0ifz>2.0513-194 xx1212(547525)0z4.992222837812nn56285212RejectH0;concludethatthefemaleshaveahighermeanverbalscore.p-value017.Population1issupplierA.Population2issupplierB.H0:0StaywithsupplierAHa:>0ChangetosupplierBRejectH0ifz>1.645()xx()(1412.5)01212z2.682222(3)(2)12nn503012p-value=.5000-.4963=.0037RejectH0;changetosupplierB.18.a.H0:0Ha:022222.52.512.36xx12nn1128412xx12069.9569.56z1.08.36xx12b.p-value=2(.5000-.3599)=.2802c.DonorejectH0.Cannotconcludethatthereisadifferencebetweenthepopulationmeanscoresforthetwogolfers.19.a.H0:0Ha:0b.t.025=2.021df=n1+n2-2=22+20-2=40RejectH0ift<-2.021orift>2.02113-195 22222nsns112211(221)(.8)(201)(1.1)c.s.9108nn2222021221111ss.9108.2948xx12nn122220xx1202.52.1t1.36s.2948xx12d.DonotrejectH0.Cannotconcludethatadifferencebetweenpopulationmeanexists.e.Withdf=40,t.05=1.684andt.10=1.303Withtwotails,p-valueisbetween.10and.20.20.a.H0:0Ha:>0b.t.05=1.711df=n1+n2-2=16+10-2=24RejectH0ift>1.71122222nsns112211(161)(.64)(101)(.75)c.s.4669nn2161021221111ss.4669.2755xx12nn121610xx1206.826.25t2.07s.2755xx12d.RejectH0.Concludethattheconsultantwiththemoreexperiencehasthehigherpopulationmeanrating.e.With24df,t.025=2.064p-valueisapproximately.02521.a.1,2,0,0,2b.dd//n551i2()dd4ic.s1dn151d.With4degreesoffreedom,t.05=2.132RejectH0ift>2.13213-196 d10dt224.sn//15dp-valueisbetween.025and.05RejectH0;concluded>0.22.a.3,-1,3,5,3,0,1b.dd//n1472i2()dd26ic.s2082.dn171d.d=2e.With6degreesoffreedomt.025=2.44722.4472.082/721.93(.07to3.93)23.Difference=ratingafter-ratingbeforeH0:d0Ha:d>0With7degreesoffreedom,rejectH0ift>1.895d=.625ands=1.3025dd.6250dt136.sn/13025./8dp-valueisgreaterthan.10DonotrejectH0;wecannotconcludethatseeingthecommercialimprovesthemeanpotentialtopurchase.24.Differences:.20,.29,.39,.02,.24,.20,.20,.52,.29,.20dd/n2.55/10.255i2()ddis.1327dn1Withdf=9,t.025=2.26213-197 sddt.025n.1327.2552.26210.255.095(.16to.35)25.Differences:8,9.5,6,10.5,15,9,11,7.5,12,5d=93.5/10=9.35ands=2.954dt.025=2.26293522622954.../ej10935211..Intervalestimateis7.24to11.4626.H0:d=0Ha:d0RejectH0ift<-2.365orift>2.365df=7Differences-.01,.03,-.06,.16,.21,.17,-.09,.11dd/n.52/8.065i2()ddis.1131dn1d0.065t1.63s.1131dn8DonotrejectH0.Cannotconcludethatthepopulationmeansdiffer.27.Usingmatchedsamples,thedifferencesareasfollows:4,-2,8,8,5,6,-4,-2,-3,0,11,-5,5,9,5H0:d0Ha:d>0d=3ands=5.21dd30dt223.sn/.521/15dp-valueisbetween.01and.02513-198 With14degreesoffreedom,rejectH0ift>1.761RejectH0.Concludethatthepopulationofreadersspendsmoretime,onaverage,watchingtelevisionthanreading.28.a.H0:1-2=0Ha:1-20Withdf=11,t.025=2.201RejectH0ift<-2.201orift>2.201Calculatethedifference,di,foreachstock.dd//.n8512708i2()ddis334.dn1xt7.34sn/dp-value0RejectH0;adecreaseinP/Eratiosisbeingprojectedfor1998.sdb.dt.025n3.347.082.201127.082.12(4.96to9.21)29.a.Difference=Pricedeluxe-PriceStandardH0:d=10Ha:d10With6degreesoffreedom,rejectH0ift<-2.447orift>2.447d=8.86ands=2.61dd88610.dt116.sn/261./7dp-valueisgreaterthan.2013-199 DonotrejectH0;wecannotrejectthehypothesisthata$10pricedifferentialexists.sdb.dt/2n2.618.862.44778.862.41(6.45to11.27)30.a.()pp=.48-.36=.1212pp()()11pp048052.(.).(.)0360641122b.s00373.pp12nn400300120.121.645(0.0373)0.120.0614(0.0586to0.1814)c.0.121.96(0.0373)0.120.0731(0.0469to0.1931)npnp200022(.)300016(.)112231.a.p0184.nn20030012F11Is(.)(.)0184081600354.pp12HGKJ200300RejectH0ifz>1.645(.22.16)0z1.69.0354RejectH0b.p-value=(.5000-.4545)=.045532.p=220/400=0.551p=192/400=0.482055045.(.).(.)048052s00353.pp1240040013-200 pp±1.96s12pp120.55-0.481.96(0.0353)0.070.0691(0.0009to0.1391)7%moreexecutivesarepredictinganincreaseinfull-timejobs.Theconfidenceintervalshowsthedifferencemaybefrom0%to14%.33.ppzs122/pp12pp()()11pp(.)(.)025075(.)(.)0160841122s0025.pp12nn496505120.25-0.16±1.96(0.25)0.09±0.05or0.04to0.1434.a.p=682/1082=.6303(63%)1p=413/1008=.4097(41%)2pp=.6303-.4097=.2206(22%)12pppp(1)(1).6303(1.6303).4097(1.4097)1122b..0213pp12nn1082100812pp1.9612p12p.22061.96(.0213).2206.0418(.1788to.2624)35.a.p=279/300=0.931p=255/300=0.852b.H0:p1-p2=0Ha:p1-p20RejectH0ifz<-1.96orifz>1.96279255p089.300300F11Is(.)(.)08901100255.pp12HGKJ300300pp0093085..12z313.s00255.pp1213-201 p-valueislessthan.001RejectH0;womenandmendifferonthisquestion.c.pp196.s12pp12(.)(.)(.)(.)093007085015s00253.pp123003000.93-0.851.96(0.0253)0.080.05(0.03to0.13)95%confident,3%to13%morewomenthanmenagreewiththisstatement.36.H0:p1p2Ha:p1>p2()pppp12b12gzspp12npnp15450675(.)16910608(.)1122p064.nn1545169112F11IF11Isp()1p(064036.)(.)0017.pp12HGKJHGnn12KJ15451691(.06750608.)0z394.0017.Since3.94>z.05=1.645,werejectH0p-value0Conclusion:Theproportionofmenthatfeelthatthedivisionofhouseworkisfairisgreaterthantheproportionofwomenthatfeelthatthedivisionofhouseworkisfair.37.H0:p1-p2=0Ha:p1-p20RejectH0ifz<-1.96orifz>1.9613-202 6360p03514.150200F11Is(.0351406486)(.)00516.pp12HGKJ150200pp63150/.04260200/.03012()pppp12bg12(.042030.)0z233.s00516.pp12p-value=2(.5000-.4901)=.0198RejectH0;thereisadifferencebetweentherecallratesforthetwocommercials.04258.().(.)030070b.(.042030.).196150200.12.10(.02to.22)npnp232(.815)210(.724)112238.p.7718nn232210121111sp(1p)(.7718)(17718).04pp12nn12232210pp120.815.724z2.28s.04pp12p-value=2(.5-.4887)=.0226p-value<.05,rejectH0.Thepopulationproportionsdiffer.NYSEisshowingagreaterproportionofstocksbelowtheir1997highs.39.H0:p1-p20Ha:p1-p20npnp240(.40)250(.32)1122p.3592nn240250121111sp(1p)(.3592)(1.3592).0434pp12nn12240250pp120.40.32z1.85s.0434pp12p-value=.5000-.4678=.032213-203 p-value<.05,rejectH0.TheproportionofusersatworkisgreaterinWashingtonD.C.22ss1240.xxz120.5nn1222(2500)(2000)40,00035,0001.64560805000646(4354to5646)41.H0:1-2=0Ha:1-20RejectH0ifz<-1.96orifz>1.96()xx()(4.13.3)01212z3.192222(2.2)(1.5)12nn12010012RejectH0;adifferenceexistswithsystemBhavingthelowermeancheckouttime.42.a.H0:1-20Ha:1-2>0RejectH0ifz>1.645b.Usingthecomputer,n1=30n2=30x=16.23x=15.7012s1=3.52s2=3.3122(3.52)(3.31)s0.88xx123030()xx0(.16231570.)12z059.s088.xx12DonotrejectH0;cannotconcludethatthemutualfundswithaloadhaveagreatermeanrateofreturn.Loadfunds16.23%;noloadfunds15.7%13-204 c.Atz=0.59,Area=0.2224p-value=0.5000-0.2224=0.277643.H0:1-2=0Ha:1-20Use25degreesoffreedom.RejectH0ift<-2.06orift>2.0622211(8)14(10)s84.1625xx121272780t1.691111284.16snn121512p-valueisbetween.10and.20DonotrejectH0;cannotconcludeadifferenceexists.44.Difference=before-afterH0:d0Ha:d>0With5degreesoffreedom,rejectH0ift>2.015d=6.167ands=6.585dd61670.dt229.sn/6585./6dp-valueisbetween.05and.10RejectH0;concludethattheprogramprovidesweightloss.45.a.Population1-1996Population2-1997H0:1-20Ha:1-2>0b.dd/./.n17414012i2()ddis033.dn113-205 Degreesoffreedom=13;t.05=1.771RejectH0ift>1.771d00.12t142.sn/033./14dp-valueisbetween.05and.10DonotrejectH0.Thesampleof14companiesshowsearningsaredowninthefourthquarterbyameanof0.12pershare.However,datadoesnotsupporttheconclusionthatmeanearningsforallcompaniesaredownin1997.46.a.H0:p1-p20Ha:p1-p2>0b.p=704/1035=.6802(68%)1p=582/1004=.5797(58%)2pp=.6802-.5797=.100512npnp1035(0.6802)1004(0.5797)1122p.6307nn10351004121111sp(1p)(.6307)(1.6307).0214pp12nn1210351004()pp0.6802.579712z4.70s.0214pp12p-value0c.RejectH0;proportionindicatinggood/excellentincreased.47.a.H0:p1-p2=0Ha:p1-p20RejectH0ifz<-1.96orifz>1.9613-206 7690p01277.400900F11Is(.0127708723)(.)002.pp12HGKJ400900pp76400/.01990900/.01012()pppp()(.019010.)01212z450.s002.pp12p-value0RejectH0;thereisadifferencebetweenclaimrates.019081.(.).(.)010090b.009196..400900.09.0432(.0468to.1332)951448.p00341.142268410F11Is(.0034109659)(.)00188.pp12HGKJ142268pp9142/.006345268/.0018712pp0063400187...0044712004470.z238.00188.p-value=2(0.5000-0.4913)=0.0174RejectH0;Thereisasignificantdifferenceindrugresistancebetweenthetwostates.NewJerseyhasthehigherdrugresistancerate.Chapter11InferencesAboutPopulationVariancesLearningObjectives1.Understandtheimportanceofvarianceinadecision-makingsituation.13-207 2Understandtheroleofstatisticalinferenceindevelopingconclusionsaboutthevarianceofasinglepopulation.223.Knowthesamplingdistributionof(n-1)s/hasachi-squaredistributionandbeabletousethisresult2todevelopaconfidenceintervalestimateof.24.Knowhowtotesthypothesesinvolving.5.Understandtheroleofstatisticalinferenceindevelopingconclusionsaboutthevariancesoftwopopulations.226.Knowthatthesamplingdistributionofs/shasanFdistributionandbeabletousethisresulttotest12hypothesesinvolvingthevariancesoftwopopulations.Solutions:1.a.11.0705b.27.4884c.9.59083d.23.2093e.9.3904613-208 22.s=2522a.With19degreesoffreedom=30.1435and=10.1170.05.9519(25)219(25)30.143510.1170215.7646.9522b.With19degreesoffreedom=32.8523and=8.90655.025.97519(25)219(25)32.85238.90655214.4653.332c.3.87.323.With15degreesoffreedom=24.9958.052RejectH0if>24.9958222(1ns)(161)(8)19.2250DonotrejectH04.a.n=182s=.3622=27.5871=8.67176(17degreesoffreedom).05.9517(.36)217(.36)27.58718.671762.22.71b..47.8422()xx25.a.s31.07n1s31.075.5722b.=16.0128=1.68987.025.975(81)(31.07)2(81)(31.07)16.01281.6898713-209 213.58128.71c.3.6911.3422()xxi6.a.s176.96n1s176.9613.3022b.=11.1433=0.484419.025.975(51)(176.96)2(51)(176.96)11.14330.484419263.521461.217.9738.2322()xxi7.a.s2.62n1s2.621.6222b.=16.0128=1.68987.025.095(81)(2.62)2(81)(2.62)16.01281.6898721.1410.85c.1.073.2922()xxi.09298.a.s.00845n1121b.s.00845.0919c.11degreesoffreedom22=21.92=3.81575.025.97522(1ns)2(1ns)22.025.975(121).008452(121).0084521.923.8157513-210 2.0042.0244d..0651.156129.H0:.00042Ha:.0004n=302=42.5569(29degreesoffreedom).052(29)(.0005)36.25.0004DonotrejectH0;theproductspecificationdoesnotappeartobeviolated.210.H0:.752Ha:.752=42.5569(29degreesoffreedom).05222(ns1)(29)(2)206.2222(.75)02Since=206.22>42.5569,rejectH0ThestandarddeviationfortelevisionsetsisgreaterthanthestandarddeviationforVCR’s.11.19degreesoffreedom22=8.90655=32.8523.975.02522RejectH0if<8.90655orif>32.8523222(ns1)(201)(.114)26.792.009216DonotrejectH0.Cannotconcludethevarianceininterestrateshaschanged.22()xxi12.s.8106n12H0:.942Ha:.9422(ns1)(11)(.8106)9.492.94013-211 2222With11degreesoffreedom,rejectif<=3.81575or>=21.92..975.0252Since=9.49isnotintherejectionregion,wecannotrejectH0.13.a.F.05=2.91b.F.025=2.76c.F.01=4.5011d.F.29.975F3.42.025,20,10RemembertoreversethedegreesoffreedomintheF.025above.14.F.05,15,19=2.23RejectH0ifF>2.232s5.81F2.422s2.4222RejectH0:conclude1215.Werecommendplacingthelargersamplevarianceinthenumerator.With=.05,F.025,20,24=2.33.RejectifF>2.33.F=8.2/4.0=2.05DonotrejectH0OrifwehadthelowertailFvalue,11F.41.025,20,24F2.41.025,24,20F=4.0/8.2=.49F>.41DonotrejectH02216.H:01222H:a12F.01,24,29=2.49RejectH0ifF>2.4922s941F2.6322s582RejectH0;Concludeadultshaveagreatervarianceinonlinetimesthanteens.217.a.Let=varianceinrepaircosts(4yearoldautomobiles)113-212 2=varianceinrepaircosts(2yearoldautomobiles)222H:01222H:a1222b.s=(170)=28,900122s=(100)=10,00022s28,9001F2.892s10,0002F.01,24,24=2.66RejectH;concludethat4yearoldautomobileshavealargervarianceinannualrepaircostscomparedto20yearoldautomobiles.Thisisexpectedduetothefactthatolderautomobilesaremorelikelytohavesomeveryexpensiverepairswhichleadtogreatervarianceintheannualrepaircosts.2218.H:01222H:a12F/2=F.025,9,6=5.5222s4.271F3.5422s2.272DonotrejectH;Cannotconcludeanydifferencebetweenvariancesofthetwoindustries.02219.H:01222H:a12F.025=2.37(Degreesoffreedomare24numerator,21denominator)UsingMinitab,Machine1:n1=25s1=.2211x1=3.328Machine1:n1=22s1=.0768x1=3.27822s(.2211)1F8.2922s(.0768)213-213 RejectH;theprocessvariancesaresignificantlydifferent.Machine1offersthebestopportunityfor0processqualityimprovements.Notethatthesamplemeansaresimilarwiththemeanbagweightsofapproximately3.3grams.However,theprocessvariancesaresignificantlydifferent.2220.H:01222H:a12F.025=2.37(Degreesoffreedomare24numerator,24denominator)With11.1thelargersamplevariance,wehaveF=11.1/2.1=5.29RejectH;thevariancesarenotequalforseniorsandmanagers.022()xix21.a.sn122s=9663.57s=19,237.73NovDec22b.H:0NovDec22H:aNovDec2s19,237.73DecF1.992s9663.57NovF.05,9,9=3.18SinceF=1.99<3.18,donotrejectH0Thereisnoevidencethatthepopulationvariancesdiffer.2222.H:0wetdry22H:awetdry2222s321024s16256wetdryF.05=2.402s1024wetF42s256drySinceF=4>2.40,rejectHandconcludethatthereisgreatervariabilityinstoppingdistancesonwet0pavement.13-214 b.Drivecarefullyonwetpavementbecauseoftheuncertaintyinstoppingdistances.2223.a.s=(30)=90022b.=30.1435and=10.1170(19degreesoffreedom).05.95(19)(900)2(19)(900)30.143510.11702567.291690.22c.23.8241.1124.With12degreesoffreedom,22=23.3367=4.40379.025.97522(12)(14.95)2(12)(14.95)23.33674.403792114.93609.0310.7224.68xi25.a.x$260.16n22()xxib.s4996.79n1s4996.7970.6922c.=32.8523=8.90655.025.975(201)(4996.78)2(201)(4996.78)32.85238.9065522889.8710,659.4553.76103.24226.a.H0:.00012Ha:.00012=21.0642(14degreesoffreedom).1013-215 22(14)(.014)27.44.00012RejectH;exceedsmaximumvariancerequirement.022b.=23.6848and=6.57063(14degreesoffreedom).05.9522(14)(.014)2(14)(.014)23.68486.570632.00012.00042227.H0:.022Ha:.022=55.7585(40degreesoffreedom).0522(40)(.16)51.2.02DonotrejectH;thevariancedoesnotappeartobeexceedingthestandard.0228.n=22s=1.52H0:2Ha:2=29.6151(21degreesoffreedom).102(21)(1.5)31.512RejectH;concludethat>1.022()xxi101.5629.s12.69n1912H:=1002Ha:1022(ns1)(8)(12.69)10.162100With8degreesoffreedom,rejectif2222<=2.73264or>=15.5073.95.0513-216 2Since=10.16isnotintherejectionregion,wecannotrejectH0.30.a.Tryn=1522=26.1190=5.62872(14degreesoffreedom).025.975(14)(64)2(14)(64)26.11905.62872234.30159.185.8612.62Asamplesizeof15wasused.b.n=25;expectedthewidthoftheintervaltobesmaller.22=39.3641=12.4011(24degreesoffreedom).05.97522(24)(8)2(24)(8)39.364112.4011239.02126.866.2511.132231.H:01222H:a12F/2=F.05,9,9=3.1822s15.81F422s7.92RejectH.ConcludethevariancesdifferwithNASDAQstocksshowingthegreatervariance.02232.H:01222H:a12F.025=1.46622s.9401F1.3922s.7972DonotrejectH;Wearenotabletoconcludestudentswhocompletethecourseandstudentswhodropout0havedifferentvariancesofgradepointaverages.13-217 233.n=16s=5.4112n=16s=2.32222H:01222H:a12F.05=2.40(Degreesoffreedomare15numerator,15denominator)2s5.41F2.352s2.32DonotrejectH;datadoesnotindicateadifferencebetweenthepopulationvariances.02234.H:01222H:a12F.05=1.94(30numeratorand24denominatordegreesoffreedom)2s251F2.082s122RejectH;concludethatthevariancesofassemblytimesarenotequal.0Chapter12TestsofGoodnessofFitandIndependenceLearningObjectives1.Knowhowtoconductagoodnessoffittest.2.Knowhowtousesampledatatotestforindependenceoftwovariables.3.Understandtheroleofthechi-squaredistributioninconductingtestsofgoodnessoffitandindependence.4.Beabletoconductagoodnessoffittestforcaseswherethepopulationishypothesizedtohaveeitheramultinomial,aPoisson,oranormalprobabilitydistribution.13-218 5.Foratestofindependence,beabletosetupacontingencytable,determinetheobservedandexpectedfrequencies,anddetermineifthetwovariablesareindependent.Solutions:1.Expectedfrequencies:e1=200(.40)=80,e2=200(.40)=80e3=200(.20)=40Actualfrequencies:f1=60,f2=120,f3=20(60-80)2(120-80)2(20-40)22=++8080404001600400=++808040=5+20+10=352=9.21034withk-1=3-1=2degreesoffreedom.012Since=35>9.21034rejectthenullhypothesis.Thepopulationproportionsarenotasstatedinthenullhypothesis.13-219 2.Expectedfrequencies:e1=300(.25)=75,e2=300(.25)=75e3=300(.25)=75,e4=300(.25)=75Actualfrequencies:f1=85,f2=95,f3=50,f4=70(85-75)2(95-75)2(50-75)2(70-75)22=+++7575757510040062525=+++757575751150=75=15.332=7.81473withk-1=4-1=3degreesoffreedom.052Since=15.33>7.81473rejectH0Weconcludethattheproportionsarenotallequal.3.H0=pABC=.29,pCBS=.28,pNBC=.25,pIND=.18Ha=TheproportionsarenotpABC=.29,pCBS=.28,pNBC=.25,pIND=.18Expectedfrequencies:300(.29)=87,300(.28)=84300(.25)=75,300(.18)=54e1=87,e2=84,e3=75,e4=54Actualfrequencies:f1=95,f2=70,f3=89,f4=462=7.81(3degreesoffreedom).0522222(95-87)(70-84)(89-75)(46-54)=+++87847554=6.87DonotrejectH0;thereisnosignificantchangeintheviewingaudienceproportions.4.ObservedExpectedHypothesizedFrequencyFrequency2CategoryProportion(fi)(ei)(fi-ei)/ei13-220 Brown0.30177151.84.18Yellow0.20135101.211.29Red0.2079101.24.87Orange0.104150.61.82Green0.103650.64.21Blue0.103850.63.14Totals:50629.512=11.07(5degreesoffreedom).05Since29.51>11.07,weconcludethatthepercentagefiguresreportedbythecompanyhavechanged.5.ObservedExpectedHypothesizedFrequencyFrequency2CategoryProportion(fi)(ei)(fi-ei)/eiFullService1/3264249.330.86Discount1/3255249.330.13Both1/3229249.331.66Totals:7482.652=4.61(2degreesoffreedom).10Since2.65<4.61,thereisnosignificantdifferenceinpreferenceamongthethreeservicechoices.6.ObservedExpectedHypothesizedFrequencyFrequency2CategoryProportion(fi)(ei)(fi-ei)/eiNewsandOpinion1/62019.17.04GeneralEditorial1/61519.17.91FamilyOriented1/63019.176.12Business/Financial1/62219.17.42FemaleOriented1/61619.17.52African-American1/61219.172.68Totals:11510.692=9.24(5degreesoffreedom).10Since10.69>9.24,weconcludethatthereisadifferenceintheproportionofadswithguiltappealsamongthesixtypesofmagazines.7.Expectedfrequencies:ei=(1/3)(135)=452222(43-45)(53-45)(39-45)=++=2.3145454513-221 2With2degreesoffreedom,=5.99.05DonotrejectH0;thereisnojustificationforconcludingadifferenceinpreferenceexists.8.H0:p1=.03,p2=.28,p3=.45,p4=.242df=3=11.34.012RejectH0if>11.342RatingObservedExpected(fi-ei)/eiExcellent24.03(400)=1212.00Good124.28(400)=1121.29Fair172.45(400)=180.36Poor80.24(400)=962.674004002=16.31RejectH0;concludethattheratingsdiffer.Acomparisonofobservedandexpectedfrequenciesshowtelephoneserviceisslightlybetterwithmoreexcellentandgoodratings.9.H0=ThecolumnvariableisindependentoftherowvariableHa=ThecolumnvariableisnotindependentoftherowvariableExpectedFrequencies:ABCP28.539.945.6Q21.530.134.42222222(20-28.5)(44-39.9)(50-45.6)(30-21.5)(26-30.1)(30-34.4)=+++++28.539.945.621.530.134.4=7.862=7.37776with(2-1)(3-1)=2degreesoffreedom.0252Since=7.86>7.37776RejectH0Concludethatthecolumnvariableisnotindependentoftherowvariable.10.H0=ThecolumnvariableisindependentoftherowvariableHa=ThecolumnvariableisnotindependentoftherowvariableExpectedFrequencies:ABCP17.500030.625021.8750Q28.750050.312535.9375R13.750024.062517.187513-222 2222(20-17.5000)(30-30.6250)(30-17.1875)=+++17.500030.625017.1875=19.782=9.48773with(3-1)(3-1)=4degreesoffreedom.052Since=19.78>9.48773RejectH0Concludethatthecolumnvariableisnotindependentofftherowvariable.11.H0:TypeofticketpurchasedisindependentofthetypeofflightHa:Typeofticketpurchasedisnotindependentofthetypeofflight.ExpectedFrequencies:e11=35.59e12=15.41e21=150.73e22=65.27e31=455.68e32=197.32ObservedExpectedFrequencyFrequency2TicketFlight(fi)(ei)(fi-ei)/eiFirstDomestic2935.591.22FirstInternational2215.412.82BusinessDomestic95150.7320.61BusinessInternational12165.2747.59FullFareDomestic518455.688.52FullFareInternational135197.3219.68Totals:920100.432=5.99with(3-1)(2-1)=2degreesoffreedom.05Since100.43>5.99,weconcludethatthetypeofticketpurchasedisnotindependentofthetypeofflight.12.a.ObservedFrequency(fij)DomesticEuropeanAsianTotalSame1255568248Different140105107352Total265160175600ExpectedFrequency(eij)DomesticEuropeanAsianTotalSame109.5366.1372.33248Different155.4793.87102.67352Total2651601756002ChiSquare(fij-eij)/eijDomesticEuropeanAsianTotalSame2.181.870.264.32Different1.541.320.183.0413-223 2=7.362Degreesoffreedom=2=5.99.05RejectH0;concludebrandloyaltyisnotindependentofmanufacturer.b.BrandLoyaltyDomestic125/265=.472(47.2%)HighestEuropean55/160=.344(34.4%)Asian68/175=.389(38.9%)13.IndustryMajorOilChemicalElectricalComputerBusiness3022.517.530Engineering3022.517.530Note:Valuesshownabovearetheexpectedfrequencies.2=11.3449(3degreesoffreedom:1x3=3).012=12.39RejectH0;concludethatmajorandindustrynotindependent.14.ExpectedFrequencies:e11=31.0e12=31.0e21=29.5e22=29.5e31=13.0e32=13.0e41=5.5e42=5.5e51=7.0e52=7.0e61=14.0e62=14.0ObservedExpectedFrequencyFrequency2MostDifficultGender(fi)(ei)(fi-ei)/eiSpouseMen3731.01.16SpouseWomen2531.01.16ParentsMen2829.50.08ParentsWomen3129.50.08ChildrenMen713.02.77ChildrenWomen1913.02.77SiblingsMen85.51.14SiblingsWomen35.51.14In-LawsMen47.01.2913-224 In-LawsWomen107.01.29OtherRelativesMen1614.00.29OtherRelativesWomen1214.00.29Totals:20013.432=11.0705with(6-1)(2-1)=5degreesoffreedom.05Since13.43>11.0705.weconcludethatgenderisnotindependentofthemostdifficultpersontobuyfor.15.ExpectedFrequencies:e11=17.16e12=12.84e21=14.88e22=11.12e31=28.03e32=20.97e41=22.31e42=16.69e51=17.16e52=12.84e61=15.45e62=11.55ObservedExpectedFrequencyFrequency2MagazineAppeal(fi)(ei)(fi-ei)/eiNewsGuilt2017.160.47NewsFear1012.840.63GeneralGuilt1514.880.00GeneralFear1111.120.00FamilyGuilt3028.030.14FamilyFear1920.970.18BusinessGuilt2222.310.00BusinessFear1716.690.01FemaleGuilt1617.160.08FemaleFear1412.840.11African-AmericanGuilt1215.450.77African-AmericanFear1511.551.03Totals:2013.412=15.09with(6-1)(2-1)=5degreesoffreedom.01Since3.41<15.09,thehypothesisofindependencecannotberejected.34.a.ObservedFrequency(fij)PharmConsumerComputerTelecomTotalCorrect207136151178672Incorrect3491228Total210140160190700ExpectedFrequency(eij)PharmConsumerComputerTelecomTotalCorrect201.6134.4153.6182.4672Incorrect8.45.66.47.628Total2101401601907002ChiSquare(fij-eij)/eij13-225 PharmConsumerComputerTelecomTotalCorrect.14.02.04.11.31Incorrect3.47.461.062.557.532=7.852Degreesoffreedom=3=7.81473.05DonotrejectH0;concludeorderfulfillmentisnotindependentofindustry.b.Thepharmaceuticalindustryisdoingthebestwith207of210(98.6%)correctlyfilledorders.17.ExpectedFrequencies:PartQualitySupplierGoodMinorDefectMajorDefectA88.766.075.14B173.0911.8310.08C133.159.107.752=7.962=9.48773(4degreesoffreedom:2x2=4).05DonotrejectH0;concludethattheassumptionofindependencecannotberejected18.ExpectedFrequencies:PartyAffiliationEducationLevelDemocraticRepublicanIndependentDidnotcompletehighschool282814Highschooldegree323216Collegedegree4040202=13.422=13.2767(4degreesoffreedom:2x2=4).01RejectH0;concludethatpartyaffiliationisnotindependentofeducationlevel.19.ExpectedFrequencies:e11=11.81e12=8.44e13=24.75e21=8.40e22=6.00e23=17.60e31=21.79e32=15.56e33=45.65ObservedExpectedFrequencyFrequency2SiskelEbert(fi)(ei)(fi-ei)/eiConCon2411.8112.57ConMixed88.440.02ConPro1324.755.58MixedCon88.400.02MixedMixed136.008.17MixedPro1117.602.48ProCon1021.796.3813-226 ProMixed915.562.77ProPro6445.657.38Totals:16045.362=13.28with(3-1)(3-1)=4degreesoffreedom.01Since45.36>13.28,weconcludethattheratingsarenotindependent.20.Firstestimatefromthesampledata.Samplesize=120.0(39)1(30)2(30)3(18)4(3)1561.3120120Therefore,weusePoissonprobabilitieswith=1.3tocomputeexpectedfrequencies.ObservedPoissonExpectedDifferencexFrequencyProbabilityFrequency(fi-ei)039.272532.7006.300130.354342.516-12.516230.230327.6362.364318.099811.9766.0244ormore3.04305.160-2.160222222(6.300)(-12.516)(2.364)(6.024)(-2.160)=++++32.70042.51627.63611.9765.160=9.03482=7.81473with5-1-1=3degreesoffreedom.052Since=9.0348>7.81473RejectH0ConcludethatthedatadonotfollowaPoissonprobabilitydistribution.221.Withn=30wewillusesixclasseswith16/3%oftheprobabilityassociatedwitheachclass.x=22.80s=6.2665Thezvaluesthatcreate6intervals,eachwithprobability.1667are-.98,-.43,0,.43,.98zCutoffvalueofx-.9822.8-.98(6.2665)=16.66-.4322.8-.43(6.2665)=20.11022.8+0(6.2665)=22.80.4322.8+.43(6.2665)=25.49.9822.8+.98(6.2665)=28.94ObservedExpectedIntervalFrequencyFrequencyDifferencelessthan16.6635-213-227 16.66-20.1175220.11-22.8055022.80-25.4975225.49-28.9435-228.94andup5502222222(2)(2)(0)(2)(2)(0)163.2055555552=9.34840with6-2-1=3degreesoffreedom.0252Since=3.209.34840DonotrejectH0Theclaimthatthedatacomesfromanormaldistributioncannotberejected.0(34)1(25)2(11)3(7)4(3)22.180UsePoissonprobabilitieswith=1.PoissonxObservedProbabilitiesExpected034.367929.432125.367929.432211.183914.71237.06134.904combineinto143.01531.224categoryof3or}moretomake5ormore-.0037.296ei5.2=4.302=5.99147(2degreesoffreedom).05DonotrejectH0;theassumptionofaPoissondistributioncannotberejected.0(15)1(31)2(20)3(15)4(13)5(4)6(2)23.2100PoissonxObservedProbabilitiesExpected015.135313.53131.270727.07220.270727.07315.180418.04413.09029.0213-228 5ormore6.05275.272=4.982=7.77944(4degreesoffreedom).10DonotrejectH0;theassumptionofaPoissondistributioncannotberejected.24.x=24.5s=3n=30Use6classesObservedExpectedIntervalFrequencyFrequencylessthan21.565521.56-23.214523.21-24.503524.50-25.797525.79-27.447527.41up452=2.82=6.25139(3degreesoffreedom:6-2-1=3).10DonotrejectH0;theassumptionofanormaldistributioncannotberejected.25.x=71s=17n=25Use5classesObservedExpectedIntervalFrequencyFrequencylessthan56.77556.7-66.57566.5-74.61574.6-84.51584.5up952=11.22=9.21034(2degreesoffreedom).01RejectH0;concludethedistributionisnotanormaldistribution.26.Observed60455936Expected505050502=8.042=7.81473(3degreesoffreedom).05RejectH0;concludethattheorderpotentialsarenotthesameineachsalesterritory.27.Observed48323791663Expected37.03306.82126.9621.1637.0313-229 2222(48–37.03)(323–306.82)(63–37.03)=+++37.03306.8237.03=41.692=13.2767(4degreesoffreedom).01Since41.69>13.2767,rejectH0.Mutualfundinvestors"attitudestowardcorporatebondsdifferfromtheirattitudestowardcorporatestock.28.Observed20204060Expected3535353522222(20–35)(20–35)(40–35)(60–35)=+++35353535=31.432=7.81473(3degreesoffreedom).05Since31.43>7.81473,rejectH0.Theparkmanagershouldnotplanonthesamenumberattendingeachday.PlanonalargerstaffforSundaysandholidays.29.Observed1316281716Expected18181818182=7.442=9.48773.05DonotrejectH0;theassumptionthatthenumberofridersisuniformlydistributedcannotberejected.30.ObservedExpectedHypothesizedFrequencyFrequency2CategoryProportion(fi)(ei)(fi-ei)/eiVerySatisfied0.281051408.75SomewhatSatisfied0.462352300.11Neither0.1255600.42SomewhatDissatisfied0.10905032.00VeryDissatisfied0.0415201.25Totals:50042.532=9.49(4degreesoffreedom).05Since42.53>9.49,weconcludethatthejobsatisfactionforcomputerprogrammersisdifferentthanthejobsatisfactionforISmanagers.13-230 31.ExpectedFrequencies:QualityShiftGoodDefective1st368.4431.562nd276.3323.673rd184.2215.782=8.112=5.99147(2degreesoffreedom).05RejectH0;concludethatshiftandqualityarenotindependent.32.ExpectedFrequencies:e11=1046.19e12=632.81e21=28.66e22=17.34e31=258.59e32=156.41e41=516.55e42=312.45ObservedExpectedFrequencyFrequency2EmploymentRegion(fi)(ei)(fi-ei)/eiFull-TimeEastern11051046.193.31Full-timeWestern574632.815.46Part-TimeEastern3128.660.19Part-TimeWestern1517.340.32Self-EmployedEastern229258.593.39Self-EmployedWestern186156.415.60NotEmployedEastern485516.551.93NotEmployedWestern344312.453.19Totals:296923.372=7.81with(4-1)(2-1)=3degreesoffreedom.05Since23.37>7.81,weconcludethatemploymentstatusisnotindependentofregion.33.Expectedfrequencies:LoanApprovalDecisionLoanOfficesApprovedRejectedMiller24.8615.14McMahon18.6411.36Games31.0718.93Runk12.437.572=2.212=7.81473(3degreesoffreedom).05DonotrejectH0;theloandecisiondoesnotappeartobedependentontheofficer.13-231 34.a.ObservedFrequency(fij)NeverMarriedMarriedDivorcedTotalMen23410610350Women21616816400Total45027426750ExpectedFrequency(eij)NeverMarriedMarriedDivorcedTotalMen210127.8712.13350Women240146.1313.87400Total450274267502ChiSquare(fij-eij)/eijNeverMarriedMarriedDivorcedTotalMen2.743.74.386.86Women2.403.27.336.002=12.862Degreesoffreedom=2=9.21.01RejectH0;concludemartialstatusisnotindependentofgender.b.MartialStatusNeverMarriedMarriedDivorcedMen66.9%30.3%2.9%Women54.0%42.0%4.0%Men100-66.9=33.1%havebeenmarriedWomen100-54.0=46.0%havebeenmarried35.ExpectedFrequencies:(50)(18)(50)(24)(50)(12)ee9,12,,e61112251001001002222(49)(1012)(46)9.7691262=9.48773(4degreesoffreedom).05Since9.76<9.48773,rejectH0.BankingtendstohavelowerP/Eratios.WecanconcludethatindustrytypeandP/Eratioarerelated.36.ExpectedFrequencies:DaysoftheWeekCountySunMonTuesWedThurFriSatTotalUrban56.747.655.156.760.172.644.2393Rural11.39.410.911.311.914.48.87813-232 Total685766687287534712=6.202=12.5916(6degreesoffreedom).05DonotrejectH0;theassumptionofindependencecannotberejected.37.x=76.83s=12.43ObservedExpectedIntervalFrequencyFrequencylessthan62.545562.54-68.503568.50-72.856572.85-76.835576.83-80.815580.81-85.167585.16-91.124591.12up552=22=11.0705(5degreesoffreedom).05DonotrejectH0;theassumptionofanormaldistributioncannotberejected.38.ExpectedFrequencies:LosAngelesSanDiegoSanFranciscoSanJoseTotalOccupied165.7124.3186.4165.7642Vacant34.325.738.634.3133Total200.0150.0225.0200.07752222(160-165.7)(116-124.3)(26-34.3)=+++165.7124.334.3=7.782=7.81473with3degreesoffreedom.052Since=7.787.81473DonotrejectH0.Wecannotconcludethatofficevacanciesaredependentonmetropolitanarea,butitisclose:thep-valueisslightlylargerthan.05.39.a.ObservedBinomialProb.ExpectedxFrequenciesn=4,p=.30Frequencies030.240124.01132.411641.16225.264626.46310.07567.5613-233 43.0081.81100100.00Theexpectedfrequencyofx=4is.81.Combinex=3andx=4intoonecategorysothatallexpectedfrequenciesare5ormore.ObservedExpectedxFrequenciesFrequencies03024.0113241.1622526.463or4138.37100100.0022b.With3degreesoffreedom,.05=7.81473.RejectH0if>7.81473.22feii6.17eiDonotrejectH0;concludethattheassumptionofabinomialdistributioncannotberejected.Chapter13AnalysisofVarianceandExperimentalDesignLearningObjectives1.Understandhowtheanalysisofvarianceprocedurecanbeusedtodetermineifthemeansofmorethantwopopulationsareequal.2.Knowtheassumptionsnecessarytousetheanalysisofvarianceprocedure.3.UnderstandtheuseoftheFdistributioninperformingtheanalysisofvarianceprocedure.4.KnowhowtosetupanANOVAtableandinterprettheentriesinthetable.5.Beabletouseoutputfromcomputersoftwarepackagestosolveanalysisofvarianceproblems.6.KnowhowtouseFisher’sleastsignificantdifference(LSD)procedureandFisher’sLSDwiththeBonferroniadjustmenttoconductstatisticalcomparisonsbetweenpairsofpopulationsmeans.7.Understandthedifferencebetweenacompletelyrandomizeddesign,arandomizedblockdesign,andfactorialexperiments.8.Knowthedefinitionofthefollowingterms:comparisonwiseTypeIerrorratepartitioning13-234 experimentwiseTypeIerrorrateblockingfactormaineffectlevelinteractiontreatmentreplication13-235 Solutions:1.a.x=(30+45+36)/3=37k2222SSTRnxxjj=5(30-37)+5(45-37)+5(36-37)=570j1MSTR=SSTR/(k-1)=570/2=285k2b.SSE(nsjj1)=4(6)+4(4)+4(6.5)=66j1MSE=SSE/(nT-k)=66/(15-3)=5.5c.F=MSTR/MSE=285/5.5=51.82F.05=3.89(2degreesoffreedomnumeratorand12denominator)SinceF=51.82>F.05=3.89,werejectthenullhypothesisthatthemeansofthethreepopulationsareequal.d.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments570228551.82Error66125.5Total636142.a.x=(153+169+158)/3=160k2222SSTRnxxjj=4(153-160)+4(169-160)+4(158-160)=536j1MSTR=SSTR/(k-1)=536/2=268k2b.SSE(nsjj1)=3(96.67)+3(97.33)+3(82.00)=828.00j1MSE=SSE/(nT-k)=828.00/(12-3)=92.00c.F=MSTR/MSE=268/92=2.91F.05=4.26(2degreesoffreedomnumeratorand9denominator)SinceF=2.91F.05=3.89werejectthenullhypothesisthatthemeansofthethreepopulationsareequal.d.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments1020251013.36Error4581238.17Total1478144.a.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments1200340080Error300605Total150063b.F.05=2.76(3degreesoffreedomnumeratorand60denominator)SinceF=80>F.05=2.76werejectthenullhypothesisthatthemeansofthe4populationsareequal.5.a.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments12026020Error216723Total33674b.F.05=3.15(2numeratordegreesoffreedomand60denominator)F.05=3.07(2numeratordegreesoffreedomand120denominator)Thecriticalvalueisbetween3.07and3.15SinceF=20mustexceedthecriticalvalue,nomatterwhatitsactualvalue,werejectthenullhypothesisthatthe3populationmeansareequal.15-237 6.Manufacturer1Manufacturer2Manufacturer3SampleMean232821SampleVariance6.674.673.33x=(23+28+21)/3=24k2222SSTRnxxjj=4(23-24)+4(28-24)+4(21-24)=104j1MSTR=SSTR/(k-1)=104/2=52k2SSE(nsjj1)=3(6.67)+3(4.67)+3(3.33)=44.01j1MSE=SSE/(nT-k)=44.01/(12-3)=4.89F=MSTR/MSE=52/4.89=10.63F.05=4.26(2degreesoffreedomnumeratorand9denominator)SinceF=10.63>F.05=4.26werejectthenullhypothesisthatthemeantimeneededtomixabatchofmaterialisthesameforeachmanufacturer.7.SuperiorPeerSubordinateSampleMean5.755.55.25SampleVariance1.642.001.93x=(5.75+5.5+5.25)/3=5.5k2222SSTRnxxjj=8(5.75-5.5)+8(5.5-5.5)+8(5.25-5.5)=1j1MSTR=SSTR/(k-1)=1/2=.5k2SSE(nsjj1)=7(1.64)+7(2.00)+7(1.93)=38.99j1MSE=SSE/(nT-k)=38.99/21=1.86F=MSTR/MSE=0.5/1.86=0.27F.05=3.47(2degreesoffreedomnumeratorand21denominator)SinceF=0.27F.05=3.68,werejectthenullhypothesisthatthemeanperceptionscoreisthesameforthethreegroupsofspecialists.9.RealEstateAgentArchitectStockbrokerSampleMean67.7361.1365.80SampleVariance117.72180.10137.12x=(67.73+61.13+65.80)/3=64.89k2222SSTRnxxjj=15(67.73-64.89)+15(61.13-64.89)+15(65.80-64.89)=345.47j1MSTR=SSTR/(k-1)=345.47/2=172.74k2SSE(nsjj1)=14(117.72)+14(180.10)+14(137.12)=6089.16j1MSE=SSE/(nT-k)=6089.16/(45-3)=144.98F=MSTR/MSE=172.74/144.98=1.19F.05=3.22(2degreesoffreedomnumeratorand42denominator)Note:Table4doesnotshowavaluefor2degreesoffreedomnumeratorand42denominator.However,thevalueof3.23correspondingto2degreesoffreedomnumeratorand40denominatorcanbeusedasanapproximation.15-239 SinceF=1.19=0.05,wecannotrejectthenullhypothesisthatthatthemeanprice/earningsratioisthesameforthesethreegroupsoffirms.111111.aLSDttMSE5.52.1792.23.23/2.025nnij55xx304515LSD;significantdifference12xx30366LSD;significantdifference13xx45369LSD;significantdifference2311b.xxtMSE12/2nn1211(3045)2.1795.5nn12-153.23=-18.23to-11.7712.a.Sample1Sample2Sample3SampleMean517758SampleVariance96.6797.3481.99x=(51+77+58)/3=62k2222SSTRnxxjj=4(51-62)+4(77-62)+4(58-62)=1,448j115-240 MSTR=SSTR/(k-1)=1,448/2=724k2SSE(nsjj1)=3(96.67)+3(97.34)+3(81.99)=828j1MSE=SSE/(nT-k)=828/(12-3)=92F=MSTR/MSE=724/92=7.87F.05=4.26(2degreesoffreedomnumeratorand9denominator)SinceF=7.87>F.05=4.26,werejectthenullhypothesisthatthemeansofthethreepopulationsareequal.1111b.LSDttMSE922.2624615.34/2.025nnij44xx517726LSD;significantdifference12xx51587LSD;nosignificantdifference13xx775819LSD;significantdifference23111113.LSDttMSE4.892.2622.453.54/2.025nn1344Sincexx232123.54,theredoesnotappeartobeanysignificantdifferencebetween13themeansofpopulation1andpopulation3.14.xxLSD1223-283.54-53.54=-8.54to-1.4615.Sincethereareonly3possiblepairwisecomparisonswewillusetheBonferroniadjustment.=.05/3=.017t.017/2=t.0085whichisapproximatelyt.01=2.6021111BSD2.602MSE2.602.51.06nnij66xx54.5.51.06;nosignificantdifference12xx5611.06;nosignificantdifference1315-241 xx4.561.51.06;significantdifference2316.a.Machine1Machine2Machine3Machine4SampleMean7.19.19.911.4SampleVariance1.21.93.701.02x=(7.1+9.1+9.9+11.4)/4=9.38k22222SSTRnxxjj=6(7.1-9.38)+6(9.1-9.38)+6(9.9-9.38)+6(11.4-9.38)=57.77j1MSTR=SSTR/(k-1)=57.77/3=19.26k2SSE(nsjj1)=5(1.21)+5(.93)+5(.70)+5(1.02)=19.30j1MSE=SSE/(nT-k)=19.30/(24-4)=.97F=MSTR/MSE=19.26/.97=19.86F.05=3.10(3degreesoffreedomnumeratorand20denominator)SinceF=19.86>F.05=3.10,werejectthenullhypothesisthatthemeantimebetweenbreakdownsisthesameforthefourmachines.b.Note:t/2isbasedupon20degreesoffreedom1111LSDttMSE0.972.086.32331.19/2.025nnij66xx9.111.42.3LSD;significantdifference2417.C=6[(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)]=.05/6=.008and/2=.004Sincethesmallestvaluefor/2inthettableis.005,wewilluset.005=2.845asanapproximationfort.004(20degreesoffreedom)11BSD2.8450.971.6266Thus,iftheabsolutevalueofthedifferencebetweenanytwosamplemeansexceeds1.62,thereissufficientevidencetorejectthehypothesisthatthecorrespondingpopulationmeansareequal.Means(1,2)(1,3)(1,4)(2,3)(2,4)(3,4)|Difference|22.84.30.82.31.5Significant?YesYesYesNoYesNo15-242 18.n1=12n2=8n3=10t/2isbasedupon27degreesoffreedomComparing1and211LSDt132.0522.70833.38.0251289.9514.754.8LSD;significantdifferenceComparing1and311LSD2.052132.0522.38333.171210|9.95-13.5|=3.55>LSD;significantdifferenceComparing2and311LSD2.052132.0522.92503.51810|14.75-13.5|=1.25F.05=3.68,werejectthehypothesisthatthemeansforthethreetreatmentsareequal.20.a.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments148827445.50Error203015135.3Total3518171111b.LSDtMSE2.131135.314.31/2nnij66|156-142|=14<14.31;nosignificantdifference|156-134|=22>14.31;significantdifference|142-134|=8<14.31;nosignificantdifference21.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments30047514.07Error160305.33Total4603422.a.H0:u1=u2=u3=u4=u5Ha:Notallthepopulationmeansareequalb.F.05=2.69(4degreesoffreedomnumeratorand30denominator)SinceF=14.07>2.69werejectH023.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments1502754.80Error2501615.63Total40018F.05=3.63(2degreesoffreedomnumeratorand16denominator)SinceF=4.80>F.05=3.63,werejectthenullhypothesisthatthemeansofthethreetreatmentsareequal.15-244 24.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments1200260043.99Error6004413.64Total180046F.05=3.23(2degreesoffreedomnumeratorand40denominator)F.05=3.15(2degreesoffreedomnumeratorand60denominator)ThecriticalFvalueisbetween3.15and3.23.SinceF=43.99exceedsthecriticalvalue,werejectthehypothesisthatthetreatmentmeansareequal.25.ABCSampleMean119107100SampleVariance146.8996.43173.788(119)10(107)10(100)x107.9328k2222SSTRnxxjj=8(119-107.93)+10(107-107.93)+10(100-107.93)=1617.9j1MSTR=SSTR/(k-1)=1617.9/2=809.95k2SSE(nsjj1)=7(146.86)+9(96.44)+9(173.78)=3,460j1MSE=SSE/(nT-k)=3,460/(28-3)=138.4F=MSTR/MSE=809.95/138.4=5.85F.05=3.39(2degreesoffreedomnumeratorand25denominator)SinceF=5.85>F.05=3.39,werejectthenullhypothesisthatthemeansofthethreetreatmentsareequal.26.a.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments4560222809.87Error624027231.11Total1080029b.F.05=3.35(2degreesoffreedomnumeratorand27denominator)SinceF=9.87>F.05=3.35,werejectthenullhypothesisthatthemeansofthethreeassemblymethodsareequal.15-245 27.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFBetween61.64320.5517.56Error23.41201.17Total85.0523F.05=3.10(3degreesoffreedomnumeratorand20denominator)SinceF=17.56>F.05=3.10,werejectthenullhypothesisthatthemeanbreakingstrengthofthefourcablesisthesame.28.506070SampleMean332928SampleVariance3217.59.5x=(33+29+28)/3=30k2222SSTRnxxjj=5(33-30)+5(29-30)+5(28-30)=70j1MSTR=SSTR/(k-1)=70/2=35k2SSE(nsjj1)=4(32)+4(17.5)+4(9.5)=236j1MSE=SSE/(nT-k)=236/(15-3)=19.67F=MSTR/MSE=35/19.67=1.78F.05=3.89(2degreesoffreedomnumeratorand12denominator)SinceF=1.78F.05=3.55,werejectthenullhypothesisthatthemeansforthethreegroupsareequal.30.Paint1Paint2Paint3Paint4SampleMean13.3139136144SampleVariance47.5.502154.5x=(133+139+136+144)/3=138k22222SSTRnxxjj=5(133-138)+5(139-138)+5(136-138)+5(144-138)=330j1MSTR=SSTR/(k-1)=330/3=110k2SSE(nsjj1)=4(47.5)+4(50)+4(21)+4(54.5)=692j1MSE=SSE/(nT-k)=692/(20-4)=43.25F=MSTR/MSE=110/43.25=2.54F.05=3.24(3degreesoffreedomnumeratorand16denominator)SinceF=2.54F.05=3.89,werejectthenullhypothesisthatthemeanmilespergallonratingsarethesameforthethreeautomobiles.32.Note:degreesoffreedomfort/2are181111LSDttMSE5.092.1011.45432.53/2.025nnij77xx17.020.43.42.53;significantdifference12xx17.025.082.53;significantdifference13xx20.4254.62.53;significantdifference2333.Note:degreesoffreedomfort/2are121111LSDttMSE22.179.81.95/2.025nnij55xx202111.95;nosignificantdifference12xx202551.95;significantdifference13xx212541.95;significantdifference2334.TreatmentMeans:x=13.6x=11.0x=10.6123BlockMeans:x=9x=7.67x=15.67x=18.67x=7.6712345OverallMean:x=176/15=11.7315-248 Step12222SSTxijx=(10-11.73)+(9-11.73)+···+(8-11.73)=354.93ijStep22222SSTRbxxj=5[(13.6-11.73)+(11.0-11.73)+(10.6-11.73)]=26.53jStep32222SSBLkxxi=3[(9-11.73)+(7.67-11.73)+(15.67-11.73)+i22(18.67-11.73)+(7.67-11.73)]=312.32Step4SSE=SST-SSTR-SSBL=354.93-26.53-312.32=16.08SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments26.53213.276.60Blocks312.32478.08Error16.0882.01Total354.9314F.05=4.46(2degreesoffreedomnumeratorand8denominator)SinceF=6.60>F.05=4.46,werejectthenullhypothesisthatthemeansofthethreetreatmentsareequal.35.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments310477.517.69Blocks85242.5Error3584.38Total43014F.05=3.84(4degreesoffreedomnumeratorand8denominator)SinceF=17.69>F.05=3.84,werejectthenullhypothesisthatthemeansofthetreatmentsareequal.36.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments900330012.60Blocks400757.14Error5002123.81Total180031F.05=3.07(3degreesoffreedomnumeratorand21denominator)15-249 SinceF=12.60>F.05=3.07,werejectthenullhypothesisthatthemeansofthetreatmentsareequal.37.TreatmentMeans:x=56x=4412BlockMeans:x=46x=49.5x=54.5123OverallMean:x=300/6=50Step12222SSTxijx=(50-50)+(42-50)+···+(46-50)=310ijStep2222SSTRbxxj=3[(56-50)+(44-50)]=216jStep32222SSBLkxxi=2[(46-50)+(49.5-50)+(54.5-50)]=73iStep4SSE=SST-SSTR-SSBL=310-216-73=21SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments216121620.57Blocks73236.5Error21210.5Total3105F.05=18.51(1degreeoffreedomnumeratorand2denominator)SinceF=20.57>F.05=18.51,werejectthenullhypothesisthatthemeantuneuptimesarethesameforbothanalyzers.38.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments45411.257.12Blocks36312Error19121.58Total1001915-250 F.05=3.26(4degreesoffreedomnumeratorand12denominator)SinceF=7.12>F.05=3.26,werejectthenullhypothesisthatthemeantotalaudittimesforthefiveauditingproceduresareequal.39.TreatmentMeans:x=16x=15x=21123BlockMeans:x=18.67x=19.33x=15.33x=14.33x=1912345OverallMean:x=260/15=17.33Step12222SSTxijx=(16-17.33)+(16-17.33)+···+(22-17.33)=175.33ijStep22222SSTRbxxj=5[(16-17.33)+(15-17.33)+(21-17.33)]=103.33jStep32222SSBLkxxi=3[(18.67-17.33)+(19.33-17.33)+···+(19-17.33)]=64.75iStep4SSE=SST-SSTR-SSBL=175.33-103.33-64.75=7.25SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatments100.33251.6756.78Blocks64.75416.19Error7.258.91Total175.3314F.05=4.46(2degreesoffreedomnumeratorand8denominator)SinceF=56.78>F.05=4.46,werejectthenullhypothesisthatthemeantimesforthethreesystemsareequal.15-251 40.TheMinitaboutputforthesedataisshownbelow:ANALYSISOFVARIANCEBPMSOURCEDFSSMSBlock92796311Treat3198056602ERROR277949294TOTAL3930550Individual95%CITreatMean----+---------+---------+---------+-------1178.0(-----*-----)2171.0(-----*----)3175.0(-----*----)4123.6(-----*----)----+---------+---------+---------+-------120.0140.0160.0180.0F.05=2.96(3degreesoffreedomnumeratorand27denominator)SinceF=6602/294=22.46>2.96,werejectthenullhypothesesthatthemeanheartrateforthefourmethodsareequal.41.FactorBFactorALevel1Level2Level3MeansLevel1x=150x=78x=84x=1041112131FactorALevel2x=110x=116x=128x=1182122232FactorBMeansx1=130x2=97x3=106x=111Step12222SSTxijkx=(135-111)+(165-111)+···+(136-111)=9,028ijkStep2222SSAbrxjx=3(2)[(104-111)+(118-111)]=588i15-252 Step32222SSBarxjx=2(2)[(130-111)+(97-111)+(106-111)]=2,328jStep4222SSABrxijxixjx=2[(150-104-130+111)+(78-104-97+111)+ij2···+(128-118-106+111)]=4,392Step5SSE=SST-SSA-SSB-SSAB=9,028-588-2,328-4,392=1,720SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFFactorA58815882.05FactorB2328211644.06Interaction4392221967.66Error17206286.67Total902811F.05=5.99(1degreeoffreedomnumeratorand6denominator)F.05=5.14(2degreesoffreedomnumeratorand6denominator)SinceF=2.05F.05=5.14,Interactionissignificant.42.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFFactorA2638.673.72FactorB23211.504.94Interaction175629.1712.52Error56242.33Total28035F.05=3.01(3degreesoffreedomnumeratorand24denominator)SinceF=3.72>F.05=3.01,FactorAissignificant.F.05=3.40(2degreesoffreedomnumeratorand24denominator)SinceF=4.94>F.05=3.40,FactorBissignificant.F.05=2.51(6degreesoffreedomnumeratorand24denominator)SinceF=12.52>F.05=2.51,Interactionissignificant15-253 43.FactorBFactorBSmallLargeMeansAx=10x=10x=1011121FactorABx=18x=28x=2321222Cx=14x=16x=1531323FactorBMeansx1=14x2=18x=16Step122222SSTxijkx=(8-16)+(12-16)+(12-16)+···+(14-16)=544ijkStep22222SSAbrxix=2(2)[(10-16)+(23-16)+(15-16)]=344iStep3222SSBarxjx=3(2)[(14-16)+(18-16)]=48jStep4222SSABrxijxixjx=2[(10-10-14+16)+···+(16-15-18+16)]=56ijStep5SSE=SST-SSA-SSB-SSAB=544-344-48-56=96SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFFactorA3442172172/16=10.75FactorB4814848/16=3.00Interaction5622828/16=1.75Error96616Total54411F.05=5.14(2degreesoffreedomnumeratorand6denominator)15-254 SinceF=10.75>F.05=5.14,FactorAissignificant,thereisadifferenceduetothetypeofadvertisementdesign.F.05=5.99(1degreeoffreedomnumeratorand6denominator)SinceF=3F.05=4.26,FactorA(gender)isnotsignificant.SinceF=13.25>F.05=3.40,FactorB(occupation)issignificant.SinceF=5.53>F.05=3.40,Interactionissignificant.46.x=(1.13+1.56+2.00)/3=1.5631x=(0.48+1.68+2.86)/3=1.6732x=(1.13+0.48)/2=0.8051x=(1.56+1.68)/2=1.6202x=(2.00+2.86)/2=2.433x=(1.13+1.56+2.00+0.48+1.68+2.86)/6=1.618Step1SST=327.50(giveninproblemstatement)Step2222SSAbrxix=3(25)[(1.563-1.618)+(1.673-1.618)]=0.4538iStep32222SSBarxjx=2(25)[(0.805-1.618)+(1.62-1.618)+(2.43-1.618)]=66.0159j15-257 Step422SSABrxijxixjx=25[(1.13-1.563-0.805+1.618)+(1.56-1.563-1.62ij22+1.618)+···+(2.86-1.673-2.43+1.618)]=14.2525Step5SSE=SST-SSA-SSB-SSAB=327.50-0.4538-66.0159-14.2525SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFFactorA0.453810.45380.2648FactorB66.1059233.008019.2608Interaction14.252527.12634.1583Error246.77781441.7137Total327.5000149F.05for1degreeoffreedomnumeratorand144degreesoffreedomdenominatorisbetween3.92and3.84.F.05for2degreesoffreedomnumeratorand144denominatorisbetween3.07and3.00.Since0.2648F.05=3.07,FactorBissignificantSince4.1583>F.05=3.07,Interactionissignificant47.a.Area1Area2SampleMean9694SampleVariance504022ss504012pooledestimate=452211estimateofstandarddeviationofxx454.741244xx969412t.424.744.74t.025=2.447(6degreesoffreedom)Sincet=.42F.05=4.26werejectthenullhypothesisthatthemeanaskingpricesforallthreeareasareequal.48.TheMinitaboutputforthesedataisshownbelow:AnalysisofVarianceSourceDFSSMSFPFactor2753.3376.618.590.000Error27546.920.3Total291300.2Individual95%CIsForMeanBasedonPooledStDevLevelNMeanStDev---------+---------+---------+-------SUV1058.6004.575(-----*-----)Small1048.8004.211(-----*----)FullSize1060.1004.701(-----*-----)15-259 ---------+---------+---------+-------PooledStDev=4.50150.055.060.0Becausethep-value=.000<=.05,wecanrejectthenullhypothesisthatthemeanresalevalueisthesame.Itappearsthatthemeanresalevalueforsmallpickuptrucksismuchsmallerthanthemeanresalevalueforsportutilityvehiclesorfull-sizepickuptrucks.49.FoodPersonalCareRetailSampleMean52.2562.2555.75SampleVariance22.2515.584.92x=(52.25+62.25+55.75)/3=56.75k2222SSTRnxxjj=4(52.25-56.75)+4(62.25-56.75)+4(55.75-56.75)=206j1MSTR=SSTR/(k-1)=206/2=103k2SSE(nsjj1)=3(22.25)+3(15.58)+3(4.92)=128.25j1MSE=SSE/(nT-k)=128.25/(12-3)=14.25F=MSTR/MSE=103/14.25=7.23F.05=4.26(2degreesoffreedomnumeratorand9denominator)SinceF=7.23exceedsthecriticalFvalue,werejectthenullhypothesisthatthemeanageofexecutivesisthesameinthethreecategoriesofcompanies.50.PhysicalCabinetSystemsLawyerTherapistMakerAnalystSampleMean50.063.769.161.2SampleVariance124.22164.68105.88136.6250.063.769.161.2x614k22222SSTRnxxjj=10(50.0-61)+10(63.7-61)+10(69.1-61)+10(61.2-61)=1939.4j1MSTR=SSTR/(k-1)=1939.4/3=646.47k2SSE(nsjj1)=9(124.22)+9(164.68)+9(105.88)+9(136.62)=4,782.60j1MSE=SSE/(nT-k)=4782.6/(40-4)=132.85F=MSTR/MSE=646.47/132.85=4.8715-260 F.05=2.84(3degreesofnumeratorand40denominator)F.05=2.76(3degreesoffreedomnumeratorand60denominator)Thus,thecriticalFvalueisbetween2.76and2.84.SinceF=4.87exceedsthecriticalFvalue,werejectthenullhypothesisthatthemeanjobsatisfactionratingisthesameforthefourprofessions.51.TheMinitaboutputforthesedataisshownbelow:AnalysisofVarianceSourceDFSSMSFPFactor2433921693.660.039Error2715991592Total2920330Individual95%CIsForMeanBasedonPooledStDevLevelNMeanStDev---+---------+---------+---------+---West10108.0023.78(-------*-------)South1091.7019.62(-------*-------)NE10121.1028.75(-------*------)---+---------+---------+---------+---PooledStDev=24.3480100120140Becausethep-value=.039<=.05,wecanrejectthenullhypothesisthatthemeanrateforthethreeregionsisthesame.52.TheMintaboutputisshownbelow:ANALYSISOFVARIANCESOURCEDFSSMSFpFACTOR31271.0423.78.740.000ERROR361744.248.4TOTAL393015.2INDIVIDUAL95PCTCI"SFORMEANBASEDONPOOLEDSTDEVLEVELNMEANSTDEV--+---------+---------+---------+----West1060.0007.218(------*-----)South1045.4007.610(------*-----)N.Cent1047.3006.778(------*-----)N.East1052.1006.152(-----*------)--+---------+---------+---------+----POOLEDSTDEV=6.96142.049.056.063.0Sincethep-value=0.000<=0.05,wecanrejectthenullhypothesisthatthatthemeanbasesalaryforartdirectorsisthesameforeachofthefourregions.15-261 53.TheMinitaboutputforthesedataisshownbelow:AnalysisofVarianceSourceDFSSMSFPFactor212.4026.2019.330.001Error3724.5960.665Total3936.998Individual95%CIsForMeanBasedonPooledStDevLevelNMeanStDev------+---------+---------+---------+Receiver157.41330.8855(-------*------)Guard136.10770.7399(-------*------)Tackle127.05830.8005(-------*-------)------+---------+---------+---------+PooledStDev=0.81536.006.607.207.80Becausethep-value=.001<=.05,wecanrejectthenullhypothesisthatthemeanratingforthethreepositionsisthesame.Itappearsthatwidereceiversandtackleshaveahighermeanratingthanguards.54.XYZSampleMean929784SampleVariance30635.33x=(92+97+44)/3=91k2222SSTRnxxjj=4(92-91)+4(97-91)+4(84-91)=344j1MSTR=SSTR/(k-1)=344/2=172k2SSE(nsjj1)=3(30)+3(6)+3(35.33)=213.99j1MSE=SSE/(nT-k)=213.99/(12-3)=23.78F=MSTR/MSE=172/23.78=7.23F.05=4.26(2degreesoffreedomnumeratorand9denominator)SinceF=7.23>F.05=4.26,werejectthenullhypothesisthatthemeanabsorbencyratingsforthethreebrandsareequal.55.FirstYearSecondYearThirdYearFourthYearSampleMean1.03-0.9915.249.81SampleVariance416.93343.04159.3155.43x=(1.03-.99+15.24+9.81)/4=6.2715-262 k22222SSTRnxxjj=7(1.03-6.27)+7(-.99-6.27)+7(15.24-6.27)+(9.81-6.27)j1=1,212.10MSTR=SSTR/(k-1)=1,212.10/3=404.03k2SSE(nsjj1)=6(416.93)+6(343.04)+6(159.31)+6(55.43)=5,848.26j1MSE=SSE/(nT-k)=5,848.26/(28-4)=243.68F=MSTR/MSE=404.03/243.68=1.66F.05=3.01(3degreesoffreedomnumeratorand24denominator)SinceF=1.66F.05=4.26,werejectthenullhypothesisthatthemeanlifetimeinhoursisthesameforthethreedesigns.59.a.NonbrowserLightBrowserHeavyBrowserSampleMean4.255.255.75SampleVariance1.071.071.36x=(4.25+5.25+5.75)/3=5.0815-264 k2222SSTRnxxjj=8(4.25-5.08)+8(5.25-5.08)+8(5.75-5.08)=9.33j1MSB=SSB/(k-1)=9.33/2=4.67k2SSW(nsjj1)=7(1.07)+7(1.07)+7(1.36)=24.5j1MSW=SSW/(nT-k)=24.5/(24-3)=1.17F=MSB/MSW=4.67/1.17=3.99F.05=3.47(2degreesoffreedomnumeratorand21denominator)SinceF=3.99>F.05=3.47,werejectthenullhypothesisthatthemeancomfortscoresarethesameforthethreegroups.1111b.LSDtMSW2.0801.171.12/2nnij88Sincetheabsolutevalueofthedifferencebetweenthesamplemeansfornonbrowsersandlightbrowsersis4.255.251,wecannotrejectthenullhypothesisthatthetwopopulationmeansareequal.60.TreatmentMeans:x=22.8x=24.8x=25.80123BlockMeans:x=19.67x=25.67x=31x=23.67x=22.3312345OverallMean:x=367/15=24.47Step12222SSTxijx=(18-24.47)+(21-24.47)+···+(24-24.47)=253.73ijStep22222SSTRbxxj=5[(22.8-24.47)+(24.8-24.47)+(25.8-24.47)]=23.33j15-265 Step32222SSBLkxxi=3[(19.67-24.47)+(25.67-24.47)+···+(22.33-24.47)]=217.02iStep4SSE=SST-SSTR-SSBL=253.73-23.33-217.02=13.38SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFTreatment23.33211.676.99Blocks217.02454.2632.49Error13.3881.67Total253.7314F.05=4.46(2degreesoffreedomnumeratorand8denominator)SinceF=6.99>F.05=4.46werejectthenullhypothesisthatthemeanmilespergallonratingsforthethreebrandsofgasolineareequal.61.IIIIIISampleMean22.824.825.8SampleVariance21.29.227.2x=(22.8+24.8+25.8)/3=24.47k2222SSTRnxxjj=5(22.8-24.47)+5(24.8-24.47)+5(25.8-24.47)=23.33j1MSTR=SSTR/(k-1)=23.33/2=11.67k2SSE(nsjj1)=4(21.2)+4(9.2)+4(27.2)=230.4j1MSE=SSE/(nT-k)=230.4/(15-3)=19.2F=MSTR/MSE=11.67/19.2=.61F.05=3.89(2degreesoffreedomnumeratorand12denominator)SinceF=.61F.05=5.14,FactorBissignificant;thatis,thereisasignificantdifferenceduetothelanguagetranslated.TypeofsystemandinteractionarenotsignificantsincebothFvaluesarelessthanthecriticalvalue.64.FactorBFactorBManualAutomaticMeansMachine1x=32x=28x=3611121FactorAMachine2x=21x=26x=23.521222FactorBMeansx1=26.5x2=27x=26.7515-268 Step12222SSTxijkx=(30-26.75)+(34-26.75)+···+(28-26.75)=151.5ijkStep2222SSAbrxix=2(2)[(30-26.75)+(23.5-26.75)]=84.5iStep3222SSBarxjx=2(2)[(26.5-26.75)+(27-26.75)]=0.5jStep4222SSABrxijxixjx=2[(30-30-26.5+26.75)+···+(28-23.5-27+26.75)]ij=40.5Step5SSE=SST-SSA-SSB-SSAB=151.5-84.5-0.5-40.5=26SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFFactorA84.5184.513FactorB.51.5.08Interaction40.5140.56.23Error2646.5Total151.57F.05=7.71(1degreeoffreedomnumeratorand4denominator)SinceF=13>F.05=7.71,FactorA(TypeofMachine)issignificant.TypeofLoadingSystemandInteractionarenotsignificantsincebothFvaluesarelessthanthecriticalvalue.Chapter14SimpleLinearRegressionLearningObjectives1.Understandhowregressionanalysiscanbeusedtodevelopanequationthatestimatesmathematicallyhowtwovariablesarerelated.2.Understandthedifferencesbetweentheregressionmodel,theregressionequation,andtheestimatedregressionequation.15-269 3.Knowhowtofitanestimatedregressionequationtoasetofsampledatabasedupontheleast-squaresmethod.4.Beabletodeterminehowgoodafitisprovidedbytheestimatedregressionequationandcomputethesamplecorrelationcoefficientfromtheregressionanalysisoutput.5.Understandtheassumptionsnecessaryforstatisticalinferenceandbeabletotestforasignificantrelationship.6.Learnhowtousearesidualplottomakeajudgementastothevalidityoftheregressionassumptions,recognizeoutliers,andidentifyinfluentialobservations.7.Knowhowtodevelopconfidenceintervalestimatesofygivenaspecificvalueofxinboththecaseofameanvalueofyandanindividualvalueofy.8.Beabletocomputethesamplecorrelationcoefficientfromtheregressionanalysisoutput.9.Knowthedefinitionofthefollowingterms:independentanddependentvariablesimplelinearregressionregressionmodelregressionequationandestimatedregressionequationscatterdiagramcoefficientofdeterminationstandarderroroftheestimateconfidenceintervalpredictionintervalresidualplotstandardizedresidualplotoutlierinfluentialobservationleverage15-270 Solutions:1a.16141210y864200123456xb.Thereappearstobealinearrelationshipbetweenxandy.c.Manydifferentstraightlinescanbedrawntoprovidealinearapproximationoftherelationshipbetweenxandy;inpartdwewilldeterminetheequationofastraightlinethat“best”representstherelationshipaccordingtotheleastsquarescriterion.d.Summationsneededtocomputetheslopeandy-interceptare:2xyx1540(x)(yyx)26(x)10iiiii()xxyy()26iib2.612()xx10ibybx82(.)()630.201yx0226..e.y02264..().10615-271 2.a.35302520y1510500246810xb.Thereappearstobealinearrelationshipbetweenxandy.c.Manydifferentstraightlinescanbedrawntoprovidealinearapproximationoftherelationshipbetweenxandy;inpartdwewilldeterminetheequationofastraightlinethat“best”representstherelationshipaccordingtotheleastsquarescriterion.d.Summationsneededtocomputetheslopeandy-interceptare:2x19y116(xxyy)()57.8(xx)30.8iiiii()xxyy()57.8iib1.876612()xx30.8ibybx232.(.)1876638(.).30331101yx3033188..e.y30331886..().190515-272 3.a.7654y32100246810xb.Summationsneededtocomputetheslopeandy-interceptare:2x26y17(xxyy)()11.6(xx)22.8iiiii()xxyy()11.6iib0.508812()xx22.8ibybx34.(.)05088(5.).20754201yx075051..c.y0750514..().27915-273 4.a.135130125120y115110105100616263646566676869xb.Thereac.Manydifferentstraightlinescanbedrawntoprovidealinearapproximationoftherelationshipbetweenxandy;inpartdwewilldeterminetheequationofastraightlinethat“best”representstherelationshipaccordingtotheleastsquarescriterion.d.Summationsneededtocomputetheslopeandy-interceptare:2x325y585(xxyy)()110(xx)20iiiii()xxyy()110iib5.512()xx20ibybx117(5.)(565)2405.01yx240555..e.yx240555..24055563..()106pounds15-274 5.a.21001900170015001300y1100900700500300100020406080100120140xb.Thereappearstobealinearrelationshipbetweenxandy.c.Manydifferentstraightlinescanbedrawntoprovidealinearapproximationoftherelationshipbetweenxandy;inpartdwewilldeterminetheequationofastraightlinethat“best”representstherelationshipaccordingtotheleastsquarescriterion.Summationsneededtocomputetheslopeandy-interceptare:2x420.6y5958.7(xxyy)()142,040.3443(xx)9847.6486iiiii()xxyy()142,040.3443iib14.423812()xx9847.6486ibybx8512429.(144238600857.)(.).154201yx15421442..d.Aonemilliondollarincreaseinmediaexpenditureswillincreasecasesalesbyapproximately14.42million.e.yx15421442..1542144270..().9939815-275 6.a.1.41.210.8y0.60.40.2066687072747678808284xb.Thereappearstobealinearrelationshipbetweenxandy.c.Summationsneededtocomputetheslopeandy-interceptare:2x667.2y7.18(xxyy)()9.0623(xx)128.7iiiii()xxyy()9.0623iib0.070412()xx128.7ibybx07978.(00704741333.)(.)602.01yx602007..d.Aonepercentincreaseinthepercentageofflightsarrivingontimewilldecreasethenumberofcomplaintsper100,000passengersby0.07.eyx602007602007....(80).04215-276 155015001450S&P1400135013009600980010000102001040010600108001100011200DJIA7.a.b.Letx=DJIAandy=S&P.Summationsneededtocomputetheslopeandy-interceptare:2x104,850y14,233(xxyy)()268,921(xx)1,806,384iiiii()xxyy()268,921iib0.1488712()xx1,806,384ibybx1423.3(.14887)(10,485)137.62901yxˆ137.630.1489c.yˆ137.630.1489(11,000)1500.27orapproximately15008.a.Summationsneededtocomputetheslopeandy-interceptare:2x121y1120.9(xxyy)()544.0429(xx)177.4286iiiii()xxyy()544.0429iib3.066312()xx177.4286ibybx1601286.(30663172857.)(.).1071301yx10713307..b.Increasingthenumberoftimesanadisairedbyonewillincreasethenumberofhouseholdexposuresbyapproximately3.07million.c.yx10713307..1071330715..().153215-277 9.a.150140130120110y100908070605002468101214xb.Summationsneededtocomputetheslopeandy-interceptare:2x70y1080(xxyy)()568(xx)142iiiii()xxyy()568iib412()xx142ibybx108()()478001yx804c.yx8048049()11615-278 9590858075OverallRating706560100150200250PerformanceScore10.a.b.Letx=performancescoreandy=overallrating.Summationsneededtocomputetheslopeandy-interceptare:2x2752y1177(xxyy)()1723.73(xx)11,867.73iiiii()xxyy()1723.73iib0.145212()xx11,867.73ibybx78.4667(.1452)(183.4667)51.8201yxˆ51.820.145c.yˆ51.820.145(225)84.4orapproximately8415-279 11.a.900.0800.0700.0600.0500.0y400.0300.0200.0100.00.00.0100.0200.0300.0400.0500.0600.0700.0800.0xb.Thereappearstobealinearrelationshipbetweenthevariables.c.Thesummationsneededtocomputetheslopeandthey-interceptare:2x2973.3y3925.6(xxyy)()453,345.042(xx)483,507.581iiiii()xxyy()453,345.042iib0.938512()xx483,507.581ibybx39256.(0938529733.)(.).1135201yx11352094..d.yx11352094..11352094..(500)5835.15-280 12.a.4000035000300002500020000Revenue150001000050000020000400006000080000100000NumberofEmployeesb.Thereappearstobeapositivelinearrelationshipbetweenthenumberofemployeesandtherevenue.c.Letx=numberofemployeesandy=revenue.Summationsneededtocomputetheslopeandy-interceptare:2x4200y1669(xxyy)()4,658,594,168(xx)14,718,343,803iiiii()xxyy()4,658,594,168iib0.31651612()xx14,718,343,803ibybx14,048(.316516)(40,299)129301yxˆ12930.3165d.yˆ1293.3165(75,000)25,03115-281 13.a.30.025.020.0y15.010.05.00.00.020.040.060.080.0100.0120.0140.0xb.Thesummationsneededtocomputetheslopeandthey-interceptare:2x399y97.1(xxyy)()1233.7(xx)7648iiiii()xxyy()1233.7iib0.1613112()xx7648ibybx1387143.(.016131)(57)467675.01yx468016..c.yx468016....468016(52.5).1308orapproximately$13,080.Theagent"srequestforanauditappearstobejustified.15-282 14.a.858075y706560607080x90100110b.Thesummationsneededtocomputetheslopeandthey-interceptare:2x1677.25y1404.3(xxyy)()897.9493(xx)3657.4568iiiii()xxyy()897.9493iib0.245512()xx3657.4568ibybx70215.(.02455)(83.8625)4963.01yx49632455..c.yx49632455..49632455..(80).693%15.a.Theestimatedregressionequationandthemeanforthedependentvariableare:yx0226..y8iiThesumofsquaresduetoerrorandthetotalsumofsquaresare22SSE(yy).1240SST()yy80iiiThus,SSR=SST-SSE=80-12.4=67.62b.r=SSR/SST=67.6/80=.845Theleastsquareslineprovidedaverygoodfit;84.5%ofthevariabilityinyhasbeenexplainedbytheleastsquaresline.c.r..845919215-283 16.a.Theestimatedregressionequationandthemeanforthedependentvariableare:yxˆ30.331.88y23.2iThesumofsquaresduetoerrorandthetotalsumofsquaresare22SSE(yyˆ)6.33SST(yy)114.80iiiThus,SSR=SST-SSE=114.80-6.33=108.472b.r=SSR/SST=108.47/114.80=.945Theleastsquareslineprovidedanexcellentfit;94.5%ofthevariabilityinyhasbeenexplainedbytheestimatedregressionequation.c.r..9459721Note:thesignforrisnegativebecausetheslopeoftheestimatedregressionequationisnegative.(b1=-1.88)17.Theestimatedregressionequationandthemeanforthedependentvariableare:yxˆ.75.51y3.4iThesumofsquaresduetoerrorandthetotalsumofsquaresare22SSE(yyˆ)5.3SST(yy)11.2iiiThus,SSR=SST-SSE=11.2-5.3=5.92r=SSR/SST=5.9/11.2=.527Weseethat52.7%ofthevariabilityinyhasbeenexplainedbytheleastsquaresline.r..527725918.a.Theestimatedregressionequationandthemeanforthedependentvariableare:yxˆ1790.5581.1y3650Thesumofsquaresduetoerrorandthetotalsumofsquaresare22SSE(yyˆ)85,135.14SST(yy)335,000iiiThus,SSR=SST-SSE=335,000-85,135.14=249,864.862b.r=SSR/SST=249,864.86/335,000=.746Weseethat74.6%ofthevariabilityinyhasbeenexplainedbytheleastsquaresline.c.r..746863715-284 19.a.Theestimatedregressionequationandthemeanforthedependentvariableare:yxˆ137.63.1489y1423.3Thesumofsquaresduetoerrorandthetotalsumofsquaresare22SSE(yyˆ)7547.14SST(yy)47,582.10iiiThus,SSR=SST-SSE=47,582.10-7547.14=40,034.962b.r=SSR/SST=40,034.96/47,582.10=.84Weseethat84%ofthevariabilityinyhasbeenexplainedbytheleastsquaresline.c.r.84.9220.a.Letx=incomeandy=homeprice.Summationsneededtocomputetheslopeandy-interceptare:2x1424y2455.5(xxyy)()4011(xx)1719.618iiiii()xxyy()4011iib2.332512()xx1719.618ibybx136.4167(2.3325)(79.1111)48.1101yxˆ48.112.3325b.Thesumofsquaresduetoerrorandthetotalsumofsquaresare22SSE(yyˆ)2017.37SST(yy)11,373.09iiiThus,SSR=SST-SSE=11,373.09–2017.37=9355.722r=SSR/SST=9355.72/11,373.09=.82Weseethat82%ofthevariabilityinyhasbeenexplainedbytheleastsquaresline.r.82.91c.yˆ48.112.3325(95)173.5orapproximately$173,50021.a.Thesummationsneededinthisproblemare:2x3450y33,700(xxyy)()712,500(xx)93,750iiiii()xxyy()712,500iib7.612()xx93,750ibybx5616.67(7.6)(575)1246.670115-285 yx12466776..b.$7.60c.Thesumofsquaresduetoerrorandthetotalsumofsquaresare:22SSE(yyˆ)233,333.33SST(yy)5,648,333.33iiiThus,SSR=SST-SSE=5,648,333.33-233,333.33=5,415,0002r=SSR/SST=5,415,000/5,648,333.33=.9587Weseethat95.87%ofthevariabilityinyhasbeenexplainedbytheestimatedregressionequation.d.yx12466776..12466776..(500)$5046.6722.a.Thesummationsneededinthisproblemare:2x613.1y70(xxyy)()5766.7(xx)45,833.9286iiiii()xxyy()5766.7iib0.125812()xx45,833.9286ibybx10(0.1258)(87.5857)1.018301yxˆ1.01830.1258b.Thesumofsquaresduetoerrorandthetotalsumofsquaresare:22SSE(yyˆ)1272.4495SST(yy)1998iiiThus,SSR=SST-SSE=1998-1272.4495=725.55052r=SSR/SST=725.5505/1998=0.3631Approximately37%ofthevariabilityinchangeinexecutivecompensationisexplainedbythetwo-yearchangeinthereturnonequity.c.r03631..06026Itreflectsalinearrelationshipthatisbetweenweakandstrong.223.a.s=MSE=SSE/(n-2)=12.4/3=4.133b.sMSE4133..20332c.()1xx0is2.033s0.643b1()210xxi15-286 b26.1d.t404.s.643b1t.025=3.182(3degreesoffreedom)Sincet=4.04>t.05=3.182werejectH0:1=0e.MSR=SSR/1=67.6F=MSR/MSE=67.6/4.133=16.36F.05=10.13(1degreeoffreedomnumeratorand3denominator)SinceF=16.36>F.05=10.13werejectH0:1=0SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFRegression67.6167.616.36Error12.434.133Total80.04224.a.s=MSE=SSE/(n-2)=6.33/3=2.11b.sMSE211..14532c.(xx)30.8is1.453s0.262b1()230.8xxib188.1d.t718.s.262b1t.025=3.182(3degreesoffreedom)Sincet=-7.18<-t.025=-3.182werejectH0:1=0e.MSR=SSR/1=8.47F=MSR/MSE=108.47/2.11=51.41F.05=10.13(1degreeoffreedomnumeratorand3denominator)SinceF=51.41>F.05=10.13werejectH0:1=0SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFRegression108.471108.4751.41Error6.3332.11Total114.804225.a.s=MSE=SSE/(n-2)=5.30/3=1.7715-287 sMSE177..1332b.(xx)22.8is1.33s0.28b1()222.8xxib.511t182.s.28b1t.025=3.182(3degreesoffreedom)Sincet=1.82t.025=2.776werejectH0:1=0b.MSR=SSR/1=249,864.86/1=249.864.86F=MSR/MSE=249,864.86/21,283.79=11.74F.05=7.71(1degreeoffreedomnumeratorand4denominator)SinceF=11.74>F.05=7.71werejectH0:1=0c.SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFRegression249864.861249864.8611.74Error85135.14421283.7915-288 Total335000527.Thesumofsquaresduetoerrorandthetotalsumofsquaresare:2SSE=(yyˆ)170SST=2442iiThus,SSR=SST-SSE=2442-170=2272MSR=SSR/1=2272SSE=SST-SSR=2442-2272=170MSE=SSE/(n-2)=170/8=21.25F=MSR/MSE=2272/21.25=106.92F.05=5.32(1degreeoffreedomnumeratorand8denominator)SinceF=106.92>F.05=5.32werejectH0:1=0.Yearsofexperienceandsalesarerelated.28.SST=411.73SSE=161.37SSR=250.36MSR=SSR/1=250.36MSE=SSE/(n-2)=161.37/13=12.413F=MSR/MSE=250.36/12.413=20.17F.05=4.67(1degreeoffreedomnumeratorand13denominator)SinceF=20.17>F.05=4.67werejectH0:1=0.29.SSE=233,333.33SST=5,648,333.33SSR=5,415,000MSE=SSE/(n-2)=233,333.33/(6-2)=58,333.33MSR=SSR/1=5,415,000F=MSR/MSE=5,415,000/58,333.25=92.83SourceofVariationSumofSquaresDegreesofFreedomMeanSquareFRegression5,415,000.0015,415,00092.83Error233,333.33458,333.33Total5,648,333.335F.05=7.71(1degreeoffreedomnumeratorand4denominator)SinceF=92.83>7.71werejectH0:1=0.Productionvolumeandtotalcostarerelated.30.UsingthecomputationsfromExercise22,SSE=1272.4495SST=1998SSR=725.550515-289 s2544899..15952()xx=45,833.9286is15.95s0.0745b1()245,833.9286xxib01258.1t169.s00745.b1t.025=2.571Sincet=1.69<2.571,wecannotrejectH0:1=0Thereisnoevidenceofasignificantrelationshipbetweenxandy.31.SST=11,373.09SSE=2017.37SSR=9355.72MSR=SSR/1=9355.72MSE=SSE/(n-2)=2017.37/16=126.0856F=MSR/MSE=9355.72/126.0856=74.20F.01=8.53(1degreeoffreedomnumeratorand16denominator)SinceF=74.20>F.01=8.53werejectH0:1=0.32.a.s=2.0332xx3(x)10i2211()xx(43)pssyˆ22.0331.11pnxx()510ib.yx0226....02264().106ytsp/2yp10.63.182(1.11)=10.63.53or7.07to14.132211()xx(43)pc.ss12.03312.32ind2nxx()510id.ytspi/2nd10.63.182(2.32)=10.67.3815-290 or3.22to17.9833.a.s=1.4532b.xx3.8(x)30.8i2211()xx(33.8)pssyˆ21.453.068pnxx()530.8iyx3033188..30331883..().2469ytsp/2yp24.693.182(.68)=24.692.16or22.53to26.852211()xx(33.8)pc.ss11.45311.61ind2nxx()530.8id.ytspi/2nd24.693.182(1.61)=24.695.12or19.57to29.8134.s=1.332xx5.2(x)22.8i221()xx1(35.2)pssyˆ21.330.85pnxx()522.8iyx075051....0750513().228ytsp/2yp2.283.182(.85)=2.282.70or-.40to4.982211()xx(35.2)pss11.3311.58ind2nxx()522.8i15-291 ytspi/2nd2.283.182(1.58)=2.285.03or-2.27to7.3135.a.s=145.892xx3.2(x)0.74i221()xx1(33.2)pssyˆ2145.8968.54pnxx()60.74iyx2905458108....29054581083().203378ytsp/2yp2,033.782.776(68.54)=2,033.78190.27or$1,843.51to$2,224.05221()xx1(33.2)pb.ss1145.891161.19ind2nxx()60.74iytspi/2nd2,033.782.776(161.19)=2,033.78447.46or$1,586.32to$2,481.2436.a.yxˆ51.819.145251.819.1452(200)80.859b.s=3.52322xx183.4667(x)11,867.73i221()xx1(200183.4667)pssyˆ23.52321.055pnxx()1511,867.73iytsp/2yp80.8592.160(1.055)=80.8592.279or78.58to83.14221()xx1(200183.4667)pc.ss13.523213.678ind2nxx()1511,867.73i15-292 ytspi/2nd80.8592.160(3.678)=80.8597.944or72.92to88.80237.a.xx57(x)7648i2s=1.88s=1.372211()xx(52.557)pssyˆ21.370.52pnxx()77648iytsp/2yp13.082.571(.52)=13.081.34or11.74to14.42or$11,740to$14,420b.sind=1.4713.082.571(1.47)=13.083.78or9.30to16.86or$9,300to$16,860c.Yes,$20,400ismuchlargerthananticipated.d.Anydeductionsexceedingthe$16,860upperlimitcouldsuggestanaudit.38.a.y12466776..(500)$5046.672b.xx575(x)93,750i2s=MSE=58,333.33s=241.52221()xx1(500575)pss1241.521267.50ind2nxx()693,750iytspi/2nd5046.674.604(267.50)=5046.671231.57or$3815.10to$6278.24c.Basedononemonth,$6000isnotoutoflinesince$3815.10to$6278.24isthepredictioninterval.However,asequenceoffivetosevenmonthswithconsistentlyhighcostsshouldcauseconcern.39.a.Summationsneededtocomputetheslopeandy-interceptare:15-293 2x227y2281.7(xxyy)()6003.41(xx)1032.1iiiii()xxyy()6003.41iib5.81669412()xx1032.1ibybx228.17(5.816694)(27.7)67.04757601yx67047658167..b.SST=39,065.14SSE=4145.141SSR=34,920.0002r=SSR/SST=34,920.000/39,065.141=0.894Theestimatedregressionequationexplained89.4%ofthevariabilityiny;averygoodfit.2c.s=MSE=4145.141/8=518.143s518.14322.76221()xx1(3527.7)pssyˆ222.768.86pnxx()101032.1iyx67047658167..6704765816735..().27063ytsp/2yp270.632.262(8.86)=270.6320.04or250.59to290.67221()xx1(3527.7)pd.ss122.76124.42ind2nxx()101032.1iytspi/2nd270.632.262(24.42)=270.6355.24or215.39to325.8740.a.9b.yˆ=20.0+7.21xc.1.3626d.SSE=SST-SSR=51,984.1-41,587.3=10,396.8MSE=10,396.8/7=1,485.315-294 F=MSR/MSE=41,587.3/1,485.3=28.00F.05=5.59(1degreeoffreedomnumeratorand7denominator)SinceF=28>F.05=5.59werejectH0:B1=0.e.yˆ=20.0+7.21(50)=380.5or$380,50041.a.yˆ=6.1092+.8951xbB.8951011b.t6.01s.149b1t.025=2.306(1degreeoffreedomnumeratorand8denominator)Sincet=6.01>t.025=2.306werejectH0:B1=0c.yˆ=6.1092+.8951(25)=28.49or$28.49permonth42a.yˆ=80.0+50.0xb.30c.F=MSR/MSE=6828.6/82.1=83.17F.05=4.20(1degreeoffreedomnumeratorand28denominator)SinceF=83.17>F.05=4.20werejectH0:B1=0.Branchofficesalesarerelatedtothesalespersons.d.yˆ=80+50(12)=680or$680,00043.a.TheMinitaboutputisshownbelow:TheregressionequationisPrice=-11.8+2.18IncomePredictorCoefSECoefTPConstant-11.8012.84-0.920.380Income2.18430.27807.860.000S=6.634R-Sq=86.1%R-Sq(adj)=84.7%AnalysisofVarianceSourceDFSSMSFPRegression12717.92717.961.750.000ResidualError10440.144.0Total113158.0PredictedValuesforNewObservations15-295 NewObsFitSEFit95.0%CI95.0%PI175.792.47(70.29,81.28)(60.02,91.56)2b.r=.861.Theleastsquareslineprovidedaverygoodfit.c.The95%confidenceintervalis70.29to81.28or$70,290to$81,280.d.The95%predictionintervalis60.02to91.56or$60,020to$91,560.44.a/b.Thescatterdiagramshowsalinearrelationshipbetweenthetwovariables.c.TheMinitaboutputisshownbelow:TheregressionequationisRental$=37.1-0.779Vacancy%PredictorCoefSECoefTPConstant37.0663.53010.500.000Vacancy%-0.77910.2226-3.500.003S=4.889R-Sq=43.4%R-Sq(adj)=39.8%AnalysisofVarianceSourceDFSSMSFPRegression1292.89292.8912.260.003ResidualError16382.3723.90Total17675.26PredictedValuesforNewObservationsNewObsFitSEFit95.0%CI95.0%PI117.592.51(12.27,22.90)(5.94,29.23)228.261.42(25.26,31.26)(17.47,39.05)ValuesofPredictorsforNewObservationsNewObsVacancy%125.0211.3d.Sincethep-value=0.003islessthan=.05,therelationshipissignificant.2e.r=.434.Theleastsquareslinedoesnotprovideaverygoodfit.f.The95%confidenceintervalis12.27to22.90or$12.27to$22.90.g.The95%predictionintervalis17.47to39.05or$17.47to$39.05.245.a.x14y76(xxyy)()200(xx)126iiiii()xxyy()200iib1.587312()xx126ibybx15.2(1.5873)(14)7.02220115-296 yxˆ7.021.59b.Theresidualsare3.48,-2.47,-4.83,-1.6,and5.22c.6420Residuals-2-4-60510152025xWithonly5observationsitisdifficulttodetermineiftheassumptionsaresatisfied.However,theplotdoessuggestcurvatureintheresidualsthatwouldindicatethattheerrortermassumptionsarenotsatisfied.Thescatterdiagramforthesedataalsoindicatesthattheunderlyingrelationshipbetweenxandymaybecurvilinear.2d.s23.782211(xx)(x14)iihi2nxx()5126iThestandardizedresidualsare1.32,-.59,-1.11,-.40,1.49.e.Thestandardizedresidualplothasthesameshapeastheoriginalresidualplot.Thecurvatureobservedindicatesthattheassumptionsregardingtheerrortermmaynotbesatisfied.46.a.yxˆ2.32.64b.15-297 43210Residuals-1-2-3-40246810xTheassumptionthatthevarianceisthesameforallvaluesofxisquestionable.Thevarianceappearstoincreaseforlargervaluesofx.47.a.Letx=advertisingexpendituresandy=revenueyxˆ29.41.55b.SST=1002SSE=310.28SSR=691.72MSR=SSR/1=691.72MSE=SSE/(n-2)=310.28/5=62.0554F=MSR/MSE=691.72/62.0554=11.15F.05=6.61(1degreeoffreedomnumeratorand5denominator)SinceF=11.15>F.05=6.61weconcludethatthetwovariablesarerelated.c.15-298 1050Residuals-5-10-152535455565PredictedValuesd.Theresidualplotleadsustoquestiontheassumptionofalinearrelationshipbetweenxandy.Eventhoughtherelationshipissignificantatthe.05levelofsignificance,itwouldbeextremelydangeroustoextrapolatebeyondtherangeofthedata.48.a.yxˆ80415-299 86420Residuals-2-4-6-802468101214xb.Theassumptionsconcerningtheerrortermappearreasonable.49.a.Letx=returnoninvestment(ROE)andy=price/earnings(P/E)ratio.yxˆ32.133.22b.21.510.50-0.5StandardizedResiduals-1-1.50102030405060xc.Thereisanunusualtrendintheresiduals.Theassumptionsconcerningtheerrortermappearquestionable.15-300 50.a.TheMINITABoutputisshownbelow:TheregressionequationisY=66.1+0.402XPredictorCoefStdevt-ratiopConstant66.1032.062.060.094X0.40230.22761.770.137s=12.62R-sq=38.5%R-sq(adj)=26.1%AnalysisofVarianceSOURCEDFSSMSFpRegression1497.2497.23.120.137Error5795.7159.1Total61292.9UnusualObservationsObs.XYFitStdev.FitResidualSt.Resid1135145.00120.424.8724.582.11RRdenotesanobs.withalargest.resid.Thestandardizedresidualsare:2.11,-1.08,.14,-.38,-.78,-.04,-.41Thefirstobservationappearstobeanoutliersinceithasalargestandardizedresidual.b.2.4+-*STDRESID---1.2+----*0.0+*--**-*--1.2+*---+---------+---------+---------+---------+---------+----YHAT110.0115.0120.0125.0130.0135.0Thestandardizedresidualplotindicatesthattheobservationx=135,y=145maybeanoutlier;notethatthisobservationhasastandardizedresidualof2.11.c.Thescatterdiagramisshownbelow-Y-*--15-301 135+--**--120+**---*-105+--*----+---------+---------+---------+---------+---------+--X105120135150165180Thescatterdiagramalsoindicatesthattheobservationx=135,y=145maybeanoutlier;theimplicationisthatforsimplelinearregressionanoutliercanbeidentifiedbylookingatthescatterdiagram.51.a.TheMinitaboutputisshownbelow:TheregressionequationisY=13.0+0.425XPredictorCoefStdevt-ratiopConstant13.0022.3965.430.002X0.42480.21162.010.091s=3.181R-sq=40.2%R-sq(adj)=30.2%AnalysisofVarianceSOURCEDFSSMSFpRegression140.7840.784.030.091Error660.7210.12Total7101.50UnusualObservationsObs.XYFitStdev.FitResidualSt.Resid712.024.0018.101.205.902.00R822.019.0022.352.78-3.35-2.16RXRdenotesanobs.withalargest.resid.Xdenotesanobs.whoseXvaluegivesitlargeinfluence.Thestandardizedresidualsare:-1.00,-.41,.01,-.48,.25,.65,-2.00,-2.16Thelasttwoobservationsinthedatasetappeartobeoutlierssincethestandardizedresidualsfortheseobservationsare2.00and-2.16,respectively.b.UsingMINITAB,weobtainedthefollowingleveragevalues:.28,.24,.16,.14,.13,.14,.14,.76MINITABidentifiesanobservationashavinghighleverageifhi>6/n;forthesedata,6/n=6/8=.75.Sincetheleveragefortheobservationx=22,y=19is.76,MINITABwouldidentifyobservation8asahighleveragepoint.Thus,weconcludethatobservation8isaninfluentialobservation.15-302 c.24.0+*-Y---20.0+*-*-*--16.0+*-*--*-12.0+*-+---------+---------+---------+---------+---------+------X0.05.010.015.020.025.0Thescatterdiagramindicatesthattheobservationx=22,y=19isaninfluentialobservation.52.a.TheMinitaboutputisshownbelow:TheregressionequationisAmount=4.09+0.196MediaExpPredictorCoefSECoefTPConstant4.0892.1681.890.096MediaExp0.195520.036355.380.001S=5.044R-Sq=78.3%R-Sq(adj)=75.6%AnalysisofVarianceSourceDFSSMSFPRegression1735.84735.8428.930.001ResidualError8203.5125.44Total9939.35UnusualObservationsObsMediaExpAmountFitSEFitResidualStResid112036.3027.553.308.752.30RRdenotesanobservationwithalargestandardizedresidualb.Minitabidentifiesobservation1ashavingalargestandardizedresidual;thus,wewouldconsiderobservation1tobeanoutlier.53.a.TheMinitaboutputisshownbelow:TheregressionequationisExposure=-8.6+7.71AiredPredictorCoefSECoefTPConstant-8.5521.65-0.390.703Aired7.71490.511915.070.00015-303 S=34.88R-Sq=96.6%R-Sq(adj)=96.2%AnalysisofVarianceSourceDFSSMSFPRegression1276434276434227.170.000ResidualError897351217Total9286169UnusualObservationsObsAiredExposureFitSEFitResidualStResid195.0758.8724.432.034.42.46RXRdenotesanobservationwithalargestandardizedresidualXdenotesanobservationwhoseXvaluegivesitlargeinfluence.b.Minitabidentifiesobservation1ashavingalargestandardizedresidual;thus,wewouldconsiderobservation1tobeanoutlier.Minitabalsoidentifiesobservation1asaninfluentialobservation.54.a.TheMinitaboutputisshownbelow:TheregressionequationisSalary=707+0.00482MktCapPredictorCoefSECoefTPConstant707.0118.05.990.000MktCap0.00481540.00080765.960.000S=379.8R-Sq=66.4%R-Sq(adj)=64.5%AnalysisofVarianceSourceDFSSMSFPRegression15129071512907135.550.000ResidualError182596647144258Total197725718UnusualObservationsObsMktCapSalaryFitSEFitResidualStResid65072173325.03149.5338.6175.51.02X17120967116.21289.586.4-1173.3-3.17RRdenotesanobservationwithalargestandardizedresidualXdenotesanobservationwhoseXvaluegivesitlargeinfluence.b.Minitabidentifiesobservation6ashavingalargestandardizedresidualandobservation17asanobservationwhosexvaluegivesitlargeinfluence.Astandardizedresidualplotagainstthepredictedvaluesisshownbelow:15-304 3210-1-2StandardizedResiduals-3-4500100015002000250030003500PredictedValues55.No.Regressionorcorrelationanalysiscanneverprovethattwovariablesarecasuallyrelated.56.Theestimateofameanvalueisanestimateoftheaverageofallyvaluesassociatedwiththesamex.Theestimateofanindividualyvalueisanestimateofonlyoneoftheyvaluesassociatedwithaparticularx.57.Todeterminewhetherornotthereisasignificantrelationshipbetweenxandy.However,ifwerejectB1=0,itdoesnotimplyagoodfit.58.a.TheMinitaboutputisshownbelow:TheregressionequationisPrice=9.26+0.711SharesPredictorCoefSECoefTPConstant9.2651.0998.430.000Shares0.71050.14744.820.001S=1.419R-Sq=74.4%R-Sq(adj)=71.2%AnalysisofVarianceSourceDFSSMSFPRegression146.78446.78423.220.001ResidualError816.1162.015Total962.900b.Sincethep-valuecorrespondingtoF=23.22=.001<=.05,therelationshipissignificant.2c.r=.744;agoodfit.Theleastsquareslineexplained74.4%ofthevariabilityinPrice.d.yˆ9.26.711(6)13.5359.a.TheMinitaboutputisshownbelow:Theregressionequationis15-305 Options=-3.83+0.296CommonPredictorCoefSECoefTPConstant-3.8345.903-0.650.529Common0.295670.0264811.170.000S=11.04R-Sq=91.9%R-Sq(adj)=91.2%AnalysisofVarianceSourceDFSSMSFPRegression11520815208124.720.000ResidualError111341122Total1216550b.yˆ3.83.296(150)40.57;approximately40.6millionsharesofoptionsgrantsoutstanding.2c.r=.919;averygoodfit.Theleastsquareslineexplained91.9%ofthevariabilityinOptions.60.a.TheMinitaboutputisshownbelow:TheregressionequationisIBM=0.275+0.950S&P500PredictorCoefStDevTPConstant0.27470.90040.310.768S&P5000.94980.35692.660.029S=2.664R-Sq=47.0%R-Sq(adj)=40.3%AnalysisofVarianceSourceDFSSMSFPRegression150.25550.2557.080.029Error856.7817.098Total9107.036b.Sincethep-value=0.029islessthan=.05,therelationshipissignificant.2c.r=.470.Theleastsquareslinedoesnotprovideaverygoodfit.d.Woolworthhashigherriskwithamarketbetaof1.25.61.a.15-306 10090807060HighTemperature5040354555657585LowTemperatureb.Itappearsthatthereisapositivelinearrelationshipbetweenthetwovariables.c.TheMinitaboutputisshownbelow:TheregressionequationisHigh=23.9+0.898LowPredictorCoefSECoefTPConstant23.8996.4813.690.002Low0.89800.11218.010.000S=5.285R-Sq=78.1%R-Sq(adj)=76.9%AnalysisofVarianceSourceDFSSMSFPRegression11792.31792.364.180.000ResidualError18502.727.9Total192294.9d.Sincethep-valuecorrespondingtoF=64.18=.000<=.05,therelationshipissignificant.2e.r=.781;agoodfit.Theleastsquareslineexplained78.1%ofthevariabilityinhightemperature.f.r.781.8862.TheMINITABoutputisshownbelow:TheregressionequationisY=10.5+0.953XPredictorCoefStdevt-ratiopConstant10.5283.7452.810.023X0.95340.13826.900.000s=4.250R-sq=85.6%R-sq(adj)=83.8%AnalysisofVarianceSOURCEDFSSMSFp15-307 Regression1860.05860.0547.620.000Error8144.4718.06Total91004.53FitStdev.Fit95%C.I.95%P.I.39.131.49(35.69,42.57)(28.74,49.52)a.yˆ=10.5+.953xb.Sincethep-valuecorrespondingtoF=47.62=.000<=.05,werejectH0:1=0.c.The95%predictionintervalis28.74to49.52or$2874to$4952d.Yes,sincetheexpectedexpenseis$3913.63.a.TheMinitaboutputisshownbelow:TheregressionequationisDefects=22.2-0.148SpeedPredictorCoefSECoefTPConstant22.1741.65313.420.000Speed-0.147830.04391-3.370.028S=1.489R-Sq=73.9%R-Sq(adj)=67.4%AnalysisofVarianceSourceDFSSMSFPRegression125.13025.13011.330.028ResidualError48.8702.217Total534.000PredictedValuesforNewObservationsNewObsFitSEFit95.0%CI95.0%PI114.7830.896(12.294,17.271)(9.957,19.608)b.Sincethep-valuecorrespondingtoF=11.33=.028<=.05,therelationshipissignificant.2c.r=.739;agoodfit.Theleastsquareslineexplained73.9%ofthevariabilityinthenumberofdefects.d.UsingtheMinitaboutputinpart(a),the95%confidenceintervalis12.294to17.271.64.a.Thereappearstobeanegativelinearrelationshipbetweendistancetoworkandnumberofdaysabsent.b.TheMINITABoutputisshownbelow:TheregressionequationisY=8.10-0.344X15-308 PredictorCoefStdevt-ratiopConstant8.09780.808810.010.000X-0.344200.07761-4.430.002s=1.289R-sq=71.1%R-sq(adj)=67.5%AnalysisofVarianceSOURCEDFSSMSFpRegression132.69932.69919.670.002Error813.3011.663Total946.000FitStdev.Fit95%C.I.95%P.I.6.3770.512(5.195,7.559)(3.176,9.577)c.Sincethep-valuecorrespondingtoF=419.67is.002<=.05.WerejectH0:1=0.2d.r=.711.Theestimatedregressionequationexplained71.1%ofthevariabilityiny;thisisareasonablygoodfit.e.The95%confidenceintervalis5.195to7.559orapproximately5.2to7.6days.65.a.LetX=theageofabusandY=theannualmaintenancecost.TheMINITABoutputisshownbelow:TheregressionequationisY=220+132XPredictorCoefStdevt-ratiopConstant220.0058.483.760.006X131.6717.807.400.000s=75.50R-sq=87.3%R-sq(adj)=85.7%AnalysisofVarianceSOURCEDFSSMSFpRegression131205031205054.750.000Error8456005700Total9357650FitStdev.Fit95%C.I.95%P.I.746.729.8(678.0,815.4)(559.5,933.9)b.Sincethep-valuecorrespondingtoF=54.75is.000<=.05,werejectH0:1=0.2c.r=.873.Theleastsquareslineprovidedaverygoodfit.d.The95%predictionintervalis559.5to933.9or$559.50to$933.9066.a.LetX=hoursspentstudyingandY=totalpointsearnedTheMINITABoutputisshownbelow:TheregressionequationisY=5.85+0.830X15-309 PredictorCoefStdevt-ratiopConstant5.8477.9720.730.484X0.82950.10957.580.000s=7.523R-sq=87.8%R-sq(adj)=86.2%AnalysisofVarianceSOURCEDFSSMSFpRegression13249.73249.757.420.000Error8452.856.6Total93702.5FitStdev.Fit95%C.I.95%P.I.84.653.67(76.19,93.11)(65.35,103.96)b.Sincethep-valuecorrespondingtoF=57.42is.000<=.05,werejectH0:1=0.c.84.65pointsd.The95%predictionintervalis65.35to103.9667.a.TheMinitaboutputisshownbelow:TheregressionequationisAudit%=-0.471+0.000039IncomePredictorCoefSECoefTPConstant-0.47100.5842-0.810.431Income0.000038680.000017312.230.038S=0.2088R-Sq=21.7%R-Sq(adj)=17.4%AnalysisofVarianceSourceDFSSMSFPRegression10.217490.217494.990.038ResidualError180.784510.04358Total191.00200PredictedValuesforNewObservationsNewObsFitSEFit95.0%CI95.0%PI10.88280.0523(0.7729,0.9927)(0.4306,1.3349)b.Sincethep-value=0.038islessthan=.05,therelationshipissignificant.2c.r=.217.Theleastsquareslinedoesnotprovideaverygoodfit.d.The95%confidenceintervalis.7729to.9927.Chapter15MultipleRegression15-310 LearningObjectives1.Understandhowmultipleregressionanalysiscanbeusedtodeveloprelationshipsinvolvingonedependentvariableandseveralindependentvariables.2.Beabletointerpretthecoefficientsinamultipleregressionanalysis.3.Knowtheassumptionsnecessarytoconductstatisticaltestsinvolvingthehypothesizedregressionmodel.4.Understandtheroleofcomputerpackagesinperformingmultipleregressionanalysis.5.Beabletointerpretandusecomputeroutputtodeveloptheestimatedregressionequation.6.Beabletodeterminehowgoodafitisprovidedbytheestimatedregressionequation.7.Beabletotestforthesignificanceoftheregressionequation.8.Understandhowmulticollinearityaffectsmultipleregressionanalysis.9.Knowhowresidualanalysiscanbeusedtomakeajudgementastotheappropriatenessofthemodel,identifyoutliers,anddeterminewhichobservationsareinfluential.15-311 Solutions:1.a.b1=.5906isanestimateofthechangeinycorrespondingtoa1unitchangeinx1whenx2isheldconstant.b2=.4980isanestimateofthechangeinycorrespondingtoa1unitchangeinx2whenx1isheldconstant.2.a.Theestimatedregressionequationisyˆ=45.06+1.94x1Anestimateofywhenx1=45isyˆ=45.06+1.94(45)=132.36b.Theestimatedregressionequationisyˆ=85.22+4.32x2Anestimateofywhenx2=15isyˆ=85.22+4.32(15)=150.02c.Theestimatedregressionequationisyˆ=-18.37+2.01x1+4.74x2Anestimateofywhenx1=45andx2=15isyˆ=-18.37+2.01(45)+4.74(15)=143.183.a.b1=3.8isanestimateofthechangeinycorrespondingtoa1unitchangeinx1whenx2,x3,andx4areheldconstant.b2=-2.3isanestimateofthechangeinycorrespondingtoa1unitchangeinx2whenx1,x3,andx4areheldconstant.b3=7.6isanestimateofthechangeinycorrespondingtoa1unitchangeinx3whenx1,x2,andx4areheldconstant.b4=2.7isanestimateofthechangeinycorrespondingtoa1unitchangeinx4whenx1,x2,andx3areheldconstant.4.a.yˆ=235+10(15)+8(10)=255;salesestimate:$255,000b.Salescanbeexpectedtoincreaseby$10foreverydollarincreaseininventoryinvestmentwhenadvertisingexpenditureisheldconstant.Salescanbeexpectedtoincreaseby$8foreverydollarincreaseinadvertisingexpenditurewheninventoryinvestmentisheldconstant.20-312 5.a.TheMinitaboutputisshownbelow:TheregressionequationisRevenue=88.6+1.60TVAdvPredictorCoefSECoefTPConstant88.6381.58256.020.000TVAdv1.60390.47783.360.015S=1.215R-Sq=65.3%R-Sq(adj)=59.5%AnalysisofVarianceSourceDFSSMSFPRegression116.64016.64011.270.015ResidualError68.8601.477Total725.500b.TheMinitaboutputisshownbelow:TheregressionequationisRevenue=83.2+2.29TVAdv+1.30NewsAdvPredictorCoefSECoefTPConstant83.2301.57452.880.000TVAdv2.29020.30417.530.001NewsAdv1.30100.32074.060.010S=0.6426R-Sq=91.9%R-Sq(adj)=88.7%AnalysisofVarianceSourceDFSSMSFPRegression223.43511.71828.380.002ResidualError52.0650.413Total725.500SourceDFSeqSSTVAdv116.640NewsAdv16.795c.No,itis1.60inpart2(a)and2.99above.Inthisexerciseitrepresentsthemarginalchangeinrevenueduetoanincreaseintelevisionadvertisingwithnewspaperadvertisingheldconstant.d.Revenue=83.2+2.29(3.5)+1.30(1.8)=$93.56or$93,5606.a.TheMinitaboutputisshownbelow:TheregressionequationisSpeed=49.8+0.0151WeightPredictorCoefSECoefTPConstant49.7819.112.610.021Weight0.0151040.0060052.520.025S=7.000R-Sq=31.1%R-Sq(adj)=26.2%13-313 AnalysisofVarianceSourceDFSSMSFPRegression1309.95309.956.330.025Error14686.0049.00Total15995.95b.TheMinitaboutputisshownbelow:TheregressionequationisSpeed=80.5-0.00312Weight+0.105HorsepwrPredictorCoefSECoefTPConstant80.4879.1398.810.000Weight-0.0031220.003481-0.900.386Horsepwr0.104710.013317.860.000S=3.027R-Sq=88.0%R-Sq(adj)=86.2%AnalysisofVarianceSourceDFSSMSFPRegression2876.80438.4047.830.000ResidualError13119.159.17Total15995.957.a.TheMinitaboutputisshownbelow:TheregressionequationisSales=66.5+0.414Compet$-0.270Heller$PredictorCoefSECoefTPConstant66.5241.881.590.156Compet$0.41390.26041.590.156Heller$-0.269780.08091-3.330.013S=18.74R-Sq=65.3%R-Sq(adj)=55.4%AnalysisofVarianceSourceDFSSMSFPRegression24618.82309.46.580.025ResidualError72457.3351.0Total97076.1b.b1=.414isanestimateofthechangeinthequantitysold(1000s)oftheHellermowerwithrespecttoa$1changeinpriceincompetitor’smowerwiththepriceoftheHellermowerheldconstant.b2=-.270isanestimateofthechangeinthequantitysold(1000s)oftheHellermowerwithrespecttoa$1changeinitspricewiththepriceofthecompetitor’smowerheldconstant.c.yˆ=66.5+0.414(170)-0.270(160)=93.68or93,680units8.a.TheMinitaboutputisshownbelow:13-314 TheregressionequationisReturn=247-32.8Safety+34.6ExpRatioPredictorCoefSECoefTPConstant247.4110.42.240.039Safety-32.8413.95-2.350.031ExpRatio34.5914.132.450.026S=16.98R-Sq=58.2%R-Sq(adj)=53.3%AnalysisofVarianceSourceDFSSMSFPRegression26823.23411.611.840.001ResidualError174899.7288.2Total1911723.0b.yˆ24732.8(7.5)34.6(2)70.29.a.TheMinitaboutputisshownbelow:Theregressionequationis%College=26.7-1.43Size+0.0757SatScorePredictorCoefSECoefTPConstant26.7151.670.520.613Size-1.42980.9931-1.440.170SatScore0.075740.039061.940.072S=12.42R-Sq=38.2%R-Sq(adj)=30.0%AnalysisofVarianceSourceDFSSMSFPRegression21430.4715.24.640.027ResidualError152312.7154.2Total173743.1b.yˆ=26.7-1.43(20)+0.0757(1000)=73.8Estimateis73.8%10.a.TheMinitaboutputisshownbelow:TheregressionequationisRevenue=33.3+7.98CarsPredictorCoefSECoefTPConstant33.3483.080.400.695Cars7.98400.632312.630.000S=226.7R-Sq=92.5%R-Sq(adj)=91.9%AnalysisofVarianceSourceDFSSMSFP13-315 Regression181920678192067159.440.000Error1366793651380Total148860003b.Anincreaseof1000carsinservicewillresultinanincreaseinrevenueof$7.98million.c.TheMinitaboutputisshownbelow:TheregressionequationisRevenue=106+8.94Cars-0.191LocationPredictorCoefSECoefTPConstant105.9785.521.240.239Cars8.94270.774611.550.000Location-0.19140.1026-1.870.087S=207.7R-Sq=94.2%R-Sq(adj)=93.2%AnalysisofVarianceSourceDFSSMSFPRegression28342186417109396.660.000Error1251781743151Total14886000311.a.SSE=SST-SSR=6,724.125-6,216.375=507.752SSR6,216.375b.R.924SST6,724.12522n1101c.RR1(1)1(1.924).902anp11021d.Theestimatedregressionequationprovidedanexcellentfit.2SSR14,052.212.a.R.926SST15,182.922n1101b.RR1(1)1(1.926).905anp11021c.Yes;afteradjustingforthenumberofindependentvariablesinthemodel,weseethat90.5%ofthevariabilityinyhasbeenaccountedfor.2SSR176013.a.R.975SST180522n1301b.RR1(1)1(1.975).971anp1304113-316 c.Theestimatedregressionequationprovidedanexcellentfit.2SSR12,00014.a.R.75SST16,00022n19b.RR1(1)1.25.68anp17c.Theadjustedcoefficientofdeterminationshowsthat68%ofthevariabilityhasbeenexplainedbythetwoindependentvariables;thus,weconcludethatthemodeldoesnotexplainalargeamountofvariability.2SSR23.43515.a.R.919SST25.522n181RR1(1)1(1.919).887anp182122b.MultipleregressionanalysisispreferredsincebothRandRshowanincreasedpercentageoftheavariabilityofyexplainedwhenbothindependentvariablesareused.16.Note:theMinitaboutputisshownwiththesolutiontoExercise6.a.No;R-Sq=31.1%b.MultipleregressionanalysisispreferredsincebothR-SqandR-Sq(adj)showanincreasedpercentageofthevariabilityofyexplainedwhenbothindependentvariablesareused.2SSR1430.417.a.R.382SST3743.122n1181RR1(1)1(1.382).30anp11821b.Thefitisnotverygood18.Note:TheMinitaboutputisshownwiththesolutiontoExercise10.a.R-Sq=94.2%R-Sq(adj)=93.2%b.Thefitisverygood.19.a.MSR=SSR/p=6,216.375/2=3,108.188SSE507.75MSE72.536np11021b.F=MSR/MSE=3,108.188/72.536=42.85F.05=4.74(2degreesoffreedomnumeratorand7denominator)13-317 SinceF=42.85>F.05=4.74theoverallmodelissignificant.c.t=.5906/.0813=7.26t.025=2.365(7degreesoffreedom)Sincet=2.365>t.025=2.365,issignificant.d.t=.4980/.0567=8.78Sincet=8.78>t.025=2.365,issignificant.20.AportionoftheMinitaboutputisshownbelow.TheregressionequationisY=-18.4+2.01X1+4.74X2PredictorCoefSECoefTPConstant-18.3717.97-1.020.341X12.01020.24718.130.000X24.73780.94845.000.002S=12.71R-Sq=92.6%R-Sq(adj)=90.4%AnalysisofVarianceSourceDFSSMSFPRegression214052.27026.143.500.000ResidualError71130.7161.5Total915182.9a.Sincethep-valuecorrespondingtoF=43.50is.000<=.05,werejectH0:==0;thereisasignificantrelationship.b.Sincethep-valuecorrespondingtot=8.13is.000<=.05,werejectH0:=0;issignificant.c.Sincethep-valuecorrespondingtot=5.00is.002<=.05,werejectH0:=0;issignificant.21.a.Inthetwoindependentvariablecasethecoefficientofx1representstheexpectedchangeinycorrespondingtoaoneunitincreaseinx1whenx2isheldconstant.Inthesingleindependentvariablecasethecoefficientofx1representstheexpectedchangeinycorrespondingtoaoneunitincreaseinx1.b.Yes.Ifx1andx2arecorrelatedonewouldexpectachangeinx1tobeaccompaniedbyachangeinx2.22.a.SSE=SST-SSR=16000-12000=40002SSE4000s571.43np--1713-318 SSR12000MSR6000p2b.F=MSR/MSE=6000/571.43=10.50F.05=4.74(2degreesoffreedomnumeratorand7denominator)SinceF=10.50>F.05=4.74,werejectH0.Thereisasignificantrelationshipamongthevariables.23.a.F=28.38F.01=13.27(2degreesoffreedom,numeratorand1denominator)SinceF>F.01=13.27,rejectH0.Alternatively,thep-valueof.002leadstothesameconclusion.b.t=7.53t.025=2.571Sincet>t.025=2.571,issignificantandx1shouldnotbedroppedfromthemodel.c.t=4.06t.025=2.571Sincet>t.025=2.571,issignificantandx2shouldnotbedroppedfromthemodel.24.Note:TheMinitaboutputisshowninpart(b)ofExercise6a.F=47.83F.05=3.81(2degreesoffreedomnumeratorand13denominator)SinceF=47.83>F.05=3.81,werejectH0:==0.Alternatively,sincethep-value=.000<=.05wecanrejectH0.b.ForWeight:H0:=0Ha:0Sincethep-value=0.386>=0.05,wecannotrejectH0ForHorsepower:H0:=0Ha:0Sincethep-value=0.000<=0.05,wecanrejectH013-319 25.a.TheMinitaboutputisshownbelow:TheregressionequationisP/E=6.04+0.692Profit%+0.265Sales%PredictorCoefSECoefTPConstant6.0384.5891.320.211Profit%0.69160.21333.240.006Sales%0.26480.18711.420.180S=5.456R-Sq=47.2%R-Sq(adj)=39.0%AnalysisofVarianceSourceDFSSMSFPRegression2345.28172.645.800.016ResidualError13387.0029.77Total15732.28b.Sincethep-value=0.016<=0.05,thereisasignificantrelationshipamongthevariables.c.ForProfit%:Sincethep-value=0.006<=0.05,Profit%issignificant.ForSales%:Sincethep-value=0.180>=0.05,Sales%isnotsignificant.26.Note:TheMinitaboutputisshownwiththesolutiontoExercise10.a.Sincethep-valuecorrespondingtoF=96.66is0.000<=.05,thereisasignificantrelationshipamongthevariables.b.ForCars:Sincethep-value=0.000<=0.05,Carsissignificantc.ForLocation:Sincethep-value=0.087>=0.05,Locationisnotsignificant27.a.yˆ=29.1270+.5906(180)+.4980(310)=289.8150b.Thepointestimateforanindividualvalueisyˆ=289.8150,thesameasthepointestimateofthemeanvalue.28.a.UsingMinitab,the95%confidenceintervalis132.16to154.16.b.UsingMinitab,the95%predictionintervalis111.13to175.18.29.a.yˆ=83.2+2.29(3.5)+1.30(1.8)=93.555or$93,555Note:InExercise5b,theMinitaboutputalsoshowsthatb0=83.230,b1=2.2902,andb2=1.3010;hence,yˆ=83.230+2.2902x1+1.3010x2.Usingthisestimatedregressionequation,weobtain13-320 yˆ=83.230+2.2902(3.5)+1.3010(1.8)=93.588or$93,588Thedifference($93,588-$93,555=$33)issimplyduetothefactthatadditionalsignificantdigitsareusedinthecomputations.Fromapracticalpointofview,however,thedifferenceisnotenoughtobeconcernedabout.Inpractice,acomputersoftwarepackageisalwaysusedtoperformthecomputationsandthiswillnotbeanissue.TheMinitaboutputisshownbelow:FitStdev.Fit95%C.I.95%P.I.93.5880.291(92.840,94.335)(91.774,95.401)NotethatthevalueofFIT(yˆ)is93.588.b.Confidenceintervalestimate:92.840to94.335or$92,840to$94,335c.Predictionintervalestimate:91.774to95.401or$91,774to$95,40130.a.Sinceweightisnotstatisticallysignificant(seeExercise24),wewilluseanestimatedregressionequationwhichusesonlyHorsepowertopredictthespeedat1/4mile.TheMinitaboutputisshownbelow:TheregressionequationisSpeed=72.6+0.0968HorsepwrPredictorCoefSECoefTPConstant72.6502.65527.360.000Horsepwr0.0967560.0098659.810.000S=3.006R-Sq=87.3%R-Sq(adj)=86.4%AnalysisofVarianceSourceDFSSMSFPRegression1869.43869.4396.210.000ResidualError14126.529.04Total15995.95UnusualObservationsObsHorsepwrSpeedFitSEFitResidualStResid2290108.000100.7090.8147.2912.52R6450116.200116.1902.0360.0100.00XRdenotesanobservationwithalargestandardizedresidualXdenotesanobservationwhoseXvaluegivesitlargeinfluence.Theoutputshowsthatthepointestimateisaspeedof101.290milesperhour.b.The95%confidenceintervalis99.490to103.089milesperhour.c.The95%predictionintervalis94.596to107.984milesperhour.31.a.UsingMinitabthe95%confidenceintervalis58.37%to75.03%.b.UsingMinitabthe95%predictionintervalis35.24%to90.59%.13-321 32.a.E(y)=+x1+x2wherex2=0iflevel1and1iflevel2b.E(y)=+x1+(0)=+x1c.E(y)=+x1+(1)=+x1+d.=E(y|level2)-E(y|level1)isthechangeinE(y)fora1unitchangeinx1holdingx2constant.33.a.twob.E(y)=+x1+x2+x3wherex2x3Level001102013c.E(y|level1)=+x1+(0)+(0)=+x1E(y|level2)=+x1+(1)+(0)=+x1+E(y|level3)=+x1+(0)+(0)=+x1+=E(y|level2)-E(y|level1)=E(y|level3)-E(y|level1)isthechangeinE(y)fora1unitchangeinx1holdingx2andx3constant.34.a.$15,300b.Estimateofsales=10.1-4.2(2)+6.8(8)+15.3(0)=56.1or$56,100c.Estimateofsales=10.1-4.2(1)+6.8(3)+15.3(1)=41.6or$41,60035.a.LetType=0ifamechanicalrepairType=1ifanelectricalrepairTheMinitaboutputisshownbelow:TheregressionequationisTime=3.45+0.617TypePredictorCoefSECoefTPConstant3.45000.54676.310.000Type0.61670.70580.870.408S=1.093R-Sq=8.7%R-Sq(adj)=0.0%13-322 AnalysisofVarianceSourceDFSSMSFPRegression10.9130.9130.760.408ResidualError89.5631.195Total910.476b.Theestimatedregressionequationdidnotprovideagoodfit.Infact,thep-valueof.408showsthattherelationshipisnotsignificantforanyreasonablevalueof.c.Person=0ifBobJonesperformedtheserviceandPerson=1ifDaveNewtonperformedtheservice.TheMinitaboutputisshownbelow:TheregressionequationisTime=4.62-1.60PersonPredictorCoefSECoefTPConstant4.62000.319214.470.000Person-1.60000.4514-3.540.008S=0.7138R-Sq=61.1%R-Sq(adj)=56.2%AnalysisofVarianceSourceDFSSMSFPRegression16.40006.400012.560.008ResidualError84.07600.5095Total910.4760d.Weseethat61.1%ofthevariabilityinrepairtimehasbeenexplainedbytherepairpersonthatperformedtheservice;anacceptable,butnotgood,fit.36.a.TheMinitaboutputisshownbelow:TheregressionequationisTime=1.86+0.291Months+1.10Type-0.609PersonPredictorCoefSECoefTPConstant1.86020.72862.550.043Months0.291440.083603.490.013Type1.10240.30333.630.011Person-0.60910.3879-1.570.167S=0.4174R-Sq=90.0%R-Sq(adj)=85.0%AnalysisofVarianceSourceDFSSMSFPRegression39.43053.143518.040.002ResidualError61.04550.1743Total910.4760b.Sincethep-valuecorrespondingtoF=18.04is.002<=.05,theoverallmodelisstatisticallysignificant.13-323 c.Thep-valuecorrespondingtot=-1.57is.167>=.05;thus,theadditionofPersonisnotstatisticallysignificant.PersonishighlycorrelatedwithMonths(thesamplecorrelationcoefficientis-.691);thus,oncetheeffectofMonthshasbeenaccountedfor,Personwillnotaddmuchtothemodel.37.a.LetPosition=0ifaguardPosition=1ifanoffensivetackle.b.TheMinitaboutputisshownbelow:TheregressionequationisRating=11.2+0.732Position+0.0222Weight-2.28SpeedPredictorCoefSECoefTPConstant11.2234.5232.480.022Position0.73240.28932.530.019Weight0.022190.010392.140.045Speed-2.27750.9290-2.450.023S=0.6936R-Sq=47.5%R-Sq(adj)=40.1%AnalysisofVarianceSourceDFSSMSFPRegression39.15623.05216.350.003ResidualError2110.10140.4810Total2419.2576c.Sincethep-valuecorrespondingtoF=6.35is.003<=.05,thereisasignificantrelationshipbetweenratingandtheindependentvariables.d.ThevalueofR-Sq(adj)is40.1%;theestimatedregressionequationdidnotprovideaverygoodfit.e.Sincethep-valueforPositionist=2.53<=.05,positionisasignificantfactorintheplayer’srating.f.yˆ11.2.732(1).0222(300)2.28(5.1)6.9638.a.TheMinitaboutputisshownbelow:TheregressionequationisRisk=-91.8+1.08Age+0.252Pressure+8.74SmokerPredictorCoefSECoefTPConstant-91.7615.22-6.030.000Age1.07670.16606.490.000Pressure0.251810.045235.570.000Smoker8.7403.0012.910.01013-324 S=5.757R-Sq=87.3%R-Sq(adj)=85.0%AnalysisofVarianceSourceDFSSMSFPRegression33660.71220.236.820.000ResidualError16530.233.1Total194190.9b.Sincethep-valuecorrespondingtot=2.91is.010<=.05,smokingisasignificantfactor.c.UsingMinitab,thepointestimateis34.27;the95%predictionintervalis21.35to47.18.Thus,theprobabilityofastroke(.2135to.4718atthe95%confidencelevel)appearstobequitehigh.ThephysicianwouldprobablyrecommendthatArtquitsmokingandbeginsometypeoftreatmentdesignedtoreducehisbloodpressure.39.a.TheMinitaboutputisshownbelow:TheregressionequationisY=0.20+2.60XPredictorCoefSECoefTPConstant0.2002.1320.090.931X2.60000.64294.040.027S=2.033R-Sq=84.5%R-Sq(adj)=79.3%AnalysisofVarianceSourceDFSSMSFPRegression167.60067.60016.350.027ResidualError312.4004.133Total480.000b.UsingMinitabweobtainedthefollowingvalues:StandardizedxiyiyˆiResidual132.8.16275.4.94358.0-1.6541110.6.2451413.2.62Thepoint(3,5)doesnotappeartofollowthetrendofremainingdata;however,thevalueofthestandardizedresidualforthispoint,-1.65,isnotlargeenoughforustoconcludethat(3,5)isanoutlier.c.UsingMinitab,weobtainedthefollowingvalues:StudentizedxiyiDeletedResidual13.1327.9135-4.42411.1913-325 514.54t.025=4.303(n-p-2=5-1-2=2degreesoffreedom)Sincethestudentizeddeletedresidualfor(3,5)is-4.42<-4.303,weconcludethatthe3rdobservationisanoutlier.40.a.TheMinitaboutputisshownbelow:TheregressionequationisY=-53.3+3.11XPredicatorCoefStdevt-ratiopConstant-53.2805.786-9.210.003X3.11000.201615.430.001s=2.851R-sq=98.8%R-sq(adj)=98.3%AnalysisofVarianceSOURCEDFSSMSFpRegression11934.41934.4238.030.001Error324.48.1Total41598.8b.UsingtheMinitabweobtainedthefollowingvalues:StudentizedxiyiDeletedResidual2212-1.942421-.1226311.792835.404070-1.90t.025=4.303(n-p-2=5-1-2=2degreesoffreedom)Sincenoneofthestudentizeddeletedresidualsarelessthan-4.303orgreaterthan4.303,noneoftheobservationscanbeclassifiedasanoutlier.c.UsingMinitabweobtainedthefollowingvalues:xiyihi2212.382421.282631.222835.204070.92Thecriticalvalueis13-326 3(p1)3(11)1.2n5Sincenoneofthevaluesexceed1.2,weconcludethattherearenoinfluentialobservationsinthedata.d.UsingMinitabweobtainedthefollowingvalues:xiyiDi2212.602421.002631.262835.03407011.09SinceD5=11.09>1(ruleofthumbcriticalvalue),weconcludethatthefifthobservationisinfluential.41.a.TheMinitaboutputappearsinthesolutiontopart(b)ofExercise5;theestimatedregressionequationis:Revenue=83.2+2.29TVAdv+1.30NewsAdvb.UsingMinitabweobtainedthefollowingvalues:StandardizedyˆResiduali96.63-1.6290.41-1.0894.341.2292.21-.3794.391.1094.24-.4094.42-1.1293.351.08Withtherelativelyfewobservations,itisdifficulttodetermineifanyoftheassumptionsregardingtheerrortermhavebeenviolated.Forinstance,anargumentcouldbemadethattheredoesnotappeartobeanypatternintheplot;alternativelyanargumentcouldbemadethatthereisacurvilinearpatternintheplot.c.Thevaluesofthestandardizedresidualsaregreaterthan-2andlessthan+2;thus,usingtest,therearenooutliers.Asafurthercheckforoutliers,weusedMinitabtocomputethefollowingstudentizeddeletedresiduals:StudentizedObservationDeletedResidual1-2.112-1.1031.314-.3351.1313-327 6-.367-1.1681.10t.025=2.776(n-p-2=8-2-2=4degreesoffreedom)Sincenoneofthestudentizeddeletedresidualsislesstan-2.776orgreaterthan2.776,weconcludethattherearenooutliersinthedata.d.UsingMinitabweobtainedthefollowingvalues:ObservationhiDi1.631.522.65.703.30.224.23.015.26.146.14.017.66.818.13.06Thecriticalaveragevalueis3(p1)3(21)1.125n8Sincenoneofthevaluesexceed1.125,weconcludethattherearenoinfluentialobservations.However,usingCook’sdistancemeasure,weseethatD1>1(ruleofthumbcriticalvalue);thus,weconcludethefirstobservationisinfluential.FinalConclusion:observations1isaninfluentialobservation.42.a.TheMinitaboutputisshownbelow:TheregressionequationisSpeed=71.3+0.107Price+0.0845HorsepwrPredictorCoefSECoefTPConstant71.3282.24831.730.000Price0.107190.039182.740.017Horsepwr0.0844960.0093069.080.000S=2.485R-Sq=91.9%R-Sq(adj)=90.7%AnalysisofVarianceSourceDFSSMSFPRegression2915.66457.8374.120.000ResidualError1380.306.18Total15995.95SourceDFSeqSSPrice1406.39Horsepwr1509.27UnusualObservations13-328 ObsPriceSpeedFitSEFitResidualStResid293.8108.000105.8822.0072.1181.45XXdenotesanobservationwhoseXvaluegivesitlargeinfluence.b.Thestandardizedresidualplotisshownbelow.Thereappearstobeaveryunusualtrendinthestandardizedresiduals.--xxx1.2+-xSRES1-x-x-x0.0+xx-x-xxx---1.2+x--x-x---------+---------+---------+---------+---------+--------FITS190.096.0102.0108.0114.0c.TheMinitaboutputshowninpart(a)didnotidentifyanyobservationswithalargestandardizedresidual;thus,theredoesnotappeartobeanyoutliersinthedata.d.TheMinitaboutputshowninpart(a)identifiesobservation2asaninfluentialobservation.43.a.TheMinitaboutputisshownbelow:Theregressionequationis%College=-26.6+0.0970SatScorePredictorCoefSECoefTPConstant-26.6137.22-0.720.485SatScore0.097030.037342.600.019S=12.83R-Sq=29.7%R-Sq(adj)=25.3%AnalysisofVarianceSourceDFSSMSFPRegression11110.81110.86.750.019ResidualError162632.3164.5Total173743.1UnusualObservationsObsSatScore%CollegeFitSEFitResidualStResid371640.0042.8610.79-2.86-0.41XXdenotesanobservationwhoseXvaluegivesitlargeinfluence.13-329 b.TheMinitaboutputshowninpartaidentifiesobservation3asaninfluentialobservation.c.TheMinitaboutputappearsinthesolutiontoExercise9;theestimatesregressionequationis%College=26.7-1.43Size+0.0757SATScored.ThefollowingMinitaboutputwasalsoprovidedaspartoftheregressionoutputforpartc.UnusualObservationsObs.Size%CollegeFitStdev.FitResidualSt.Resid330.040.038.0410.971.960.34XXdenotesanobs.whoseXvaluegivesitlargeinfluence.Observation3isstillidentifiedasaninfluentialobservation.44.a.Theexpectedincreaseinfinalcollegegradepointaveragecorrespondingtoaonepointincreaseinhighschoolgradepointaverageis.0235whenSATmathematicsscoredoesnotchange.Similarly,theexpectedincreaseinfinalcollegegradepointaveragecorrespondingtoaonepointincreaseintheSATmathematicsscoreis.00486whenthehighschoolgradepointaveragedoesnotchange.b.yˆ=-1.41+.0235(84)+.00486(540)=3.1945.a.Jobsatisfactioncanbeexpectedtodecreaseby8.69unitswithaoneunitincreaseinlengthofserviceifthewageratedoesnotchange.Adollarincreaseinthewagerateisassociatedwitha13.5pointincreaseinthejobsatisfactionscorewhenthelengthofservicedoesnotchange.b.yˆ=14.4-8.69(4)+13.5(6.5)=67.3946.a.Thecomputeroutputwiththemissingvaluesfilledinisasfollows:TheregressionequationisY=8.103+7.602X1+3.111X2PredicatorCoefStdevt-ratioConstant8.1032.6673.04X17.6022.1053.61X23.1110.6135.08s=3.35R-sq=92.3%R-sq(adj)=91.0%AnalysisofVarianceSOURCEDFSSMSFRegression2161280671.82Error12134,6711.2225Total141746.67b.t.025=2.179(12DF)for:3.61>2.179;rejectH0:=0for:5.08>2.179;rejectH0:=013-330 c.Seecomputeroutput.214d.R1(1.923).91a1247.a.TheregressionequationisY=-1.41+.0235X1+.00486X2PredictorCoefStdevt-ratioConstant-1.40530.4848-2.90X10.0234670.0086662.71X2.004860.0010774.51s=0.1298R-sq=93.7%R-sq(adj)=91.9%AnalysisofVarianceSOURCEDFSSMSFRegression21.76209.88152.44Error7.1179.0168Total91.88000b.F.05=4.74(2DFnumerator,7DFdenominator)F=52.44>F.05;significantrelationship.2SSRc.R.937SST29R1(1.937).919a7goodfitd.t.025=2.365(7DF)forB:t=2.71>2.365;rejectH0:B=0forB:t=4.51>2.365;rejectH0:B=048.a.TheregressionequationisY=14.4-8.69X1+13.52X2PredictorCoefStdevt-ratioConstant14.4488.1911.76X1-8.691.555-5.59X213.5172.0856.48s=3.773R-sq=90.1%R-sq(adj)=86.1%AnalysisofVariance13-331 SOURCEDFSSMSFRegression2648.83324.41522.79Error571.1714.234Total7720.00b.F.05=5.79(5DF)F=22.79>F.05;significantrelationship.2SSRc.R.901SST27R1(1.901).861a5goodfitd.t.025=2.571(5DF)for:t=-5.59<-2.571;rejectH0:=0for:t=6.48>2.571;rejectH0:=049.a.TheMinitaboutputisshownbelow:TheregressionequationisPrice=12.8+2.26BookValPredictorCoefSECoefTPConstant12.7936.6241.930.064BookVal2.26490.66313.420.002S=19.50R-Sq=29.4%R-Sq(adj)=26.9%AnalysisofVarianceSourceDFSSMSFPRegression14433.94433.911.670.002Error2810642.3380.1Total2915076.1b.ThevalueofR-sqis29.4%;theestimatedregressionequationdoesnotprovideagoodfit.c.TheMinitaboutputisshownbelow:TheregressionequationisPrice=5.88+2.54BookVal+0.484ReturnEqPredictorCoefSECoefTPConstant5.8775.5451.060.299BookVal2.53560.53314.760.000ReturnEq0.48410.11744.120.00013-332 S=15.55R-Sq=56.7%R-Sq(adj)=53.5%AnalysisofVarianceSourceDFSSMSFPRegression28544.24272.117.660.000Error276531.9241.9Total2915076.1Sincethep-valuecorrespondingtotheFtestis0.000,therelationshipissignificant.50.a.TheMinitaboutputisshownbelow:TheregressionequationisSpeed=97.6+0.0693Price-0.00082Weight+0.0590Horsepwr-2.48Zero60PredictorCoefSECoefTPConstant97.5711.798.270.000Price0.069280.038051.820.096Weight-0.0008160.002593-0.310.759Horsepwr0.059010.015433.820.003Zero60-2.48360.9601-2.590.025S=2.127R-Sq=95.0%R-Sq(adj)=93.2%AnalysisofVarianceSourceDFSSMSFPRegression4946.18236.5552.280.000ResidualError1149.774.52Total15995.95b.Sincethep-valuecorrespondingtotheFtestis0.000,therelationshipissignificant.c.Sincethep-valuescorrespondingtothettestforbothHorsepwr(p-value=.003)andZero60(p-value=.025)arelessthan.05,bothoftheseindependentvariablesaresignificant.d.TheMinitaboutputisshownbelow:TheregressionequationisSpeed=103+0.0558Horsepwr-3.19Zero60PredictorCoefSECoefTPConstant103.1039.44810.910.000Horsepwr0.055820.014523.840.002Zero60-3.18760.9658-3.300.006S=2.301R-Sq=93.1%R-Sq(adj)=92.0%AnalysisofVariance13-333 SourceDFSSMSFPRegression2927.12463.5687.540.000ResidualError1368.845.30Total15995.95SourceDFSeqSSHorsepwr1869.43Zero60157.68UnusualObservationsObsHorsepwrSpeedFitSEFitResidualStResid2290108.000103.3521.0154.6482.25R1215584.60082.7471.7731.8531.26XRdenotesanobservationwithalargestandardizedresidualXdenotesanobservationwhoseXvaluegivesitlargeinfluence.e.Thestandardizedresidualplotisshownbelow:-SRES-x--1.5+-xx---2xx0.0+xx2-x--xx--1.5+-xx-----+---------+---------+---------+---------+---------+--FIT84.090.096.0102.0108.0114.0Thereisanunusualtrendintheplotandoneobservationappearstobeanoutlier.f.TheMinitaboutputindicatesthatobservation2isanoutlierg.TheMinitaboutputindicatesthatobservation12isaninfluentialobservation.51.a.TheMinitaboutputisshownbelow:640+-xExposure---480+-x13-334 ---x320+----x160+x3x-x------+---------+---------+---------+---------+---------+TimesAir153045607590b.TheMinitaboutputisshownbelow:TheregressionequationisExposure=53.2+6.74TimesAirPredictorCoefSECoefTPConstant53.2416.533.220.012TimesAir6.74270.447215.080.000S=31.70R-Sq=96.6%R-Sq(adj)=96.2%AnalysisofVarianceSourceDFSSMSFPRegression1228520228520227.360.000Error880411005Total9236561Sincethep-valueis0.000,therelationshipissignificant.c.TheMinitaboutputisshownbelow:TheregressionequationisExposure=73.1+5.04TimesAir+101BigAdsPredictorCoefSECoefTPConstant73.0637.5079.730.000TimesAir5.03680.326815.410.000BigAds101.1115.996.320.000S=13.08R-Sq=99.5%R-Sq(adj)=99.3%AnalysisofVarianceSourceDFSSMSFPRegression2235363117682687.840.000Error71198171Total923656113-335 d.Thep-valuecorrespondingtothettestforBigAdsis0.000;thus,thedummyvariableissignificant.e.Thedummyvariableenablesustofittwodifferentlinestothedata;thisapproachisreferredtoaspiecewiselinearapproximation.52.a.TheMinitaboutputisshownbelow:Resale%=38.8+0.000766PricePredictorCoefSECoefTPConstant38.7724.3488.920.000Price0.00076560.00019004.030.000S=5.421R-Sq=36.7%R-Sq(adj)=34.4%AnalysisofVarianceSourceDFSSMSFPRegression1477.25477.2516.240.000ResidualError28822.9229.39Total291300.17Sincethep-valuecorrespondingtoF=16.24is.000<=.05,thereisasignificantrelationshipbetweenResale%andPrice.b.R-Sq=36.7%;notaverygoodfit.c.LetType1=0andType2=0ifasmallpickup;Type1=1andType2=0ifafull-sizepickup;andType1=0andType2=1ifasportutility.TheMinitaboutputusingType1,Type2,andPriceisshownbelow:TheregressionequationisResale%=42.6+9.09Type1+7.92Type2+0.000341PricePredictorCoefSECoefTPConstant42.5543.56211.950.000Type19.0902.2484.040.000Type27.9172.1633.660.001Price0.00034150.00018001.900.069S=4.298R-Sq=63.1%R-Sq(adj)=58.8%AnalysisofVarianceSourceDFSSMSFPRegression3819.77273.2614.790.000ResidualError26480.4018.48Total291300.17d.Sincethep-valuecorrespondingtoF=14.79is.000<=.05,thereisasignificantrelationshipbetweenResale%andtheindependentvariables.Notethatindividually,Priceisnotsignificantatthe.05levelofsignificance.IfwereruntheregressionusingjustType1andType2thevalueofR-Sq(adj)decreasesto54.4%,adropofonly4%.Thus,itappearsthatforthesedata,thetypeofvehicleisthestrongestpredictoroftheresalevalue.Chapter1613-336 RegressionAnalysis:ModelBuildingLearningObjectives1.Learnhowthegenerallinearmodelcanbeusedtomodelproblemsinvolvingcurvilinearrelationships.2.Understandtheconceptofinteractionandhowitcanbeaccountedforinthegenerallinearmodel.3.UnderstandhowanFtestcanbeusedtodeterminewhentoaddordeleteoneormorevariables.4.Developanappreciationforthecomplexitiesinvolvedinsolvinglargerregressionanalysisproblems.5.Understandhowvariableselectionprocedurescanbeusedtochooseasetofindependentvariablesforanestimatedregressionequation.6.KnowhowtheDurban-Watsontestcanbeusedtotestforautocorrelation.7.Learnhowanalysisofvarianceandexperimentaldesignproblemscanbeanalyzedusingaregressionmodel.Solutions:13-337 1.a.TheMinitaboutputisshownbelow:TheregressionequationisY=-6.8+1.23XPredictorCoefStdevt-ratiopConstant-6.7714.17-0.480.658X1.22960.46972.620.059s=7.269R-sq=63.1%R-sq(adj)=53.9%AnalysisofVarianceSOURCEDFSSMSFpRegression1362.13362.136.850.059Error4211.3752.84Total5573.50b.Sincethep-valuecorrespondingtoF=6.85is0.59>therelationshipisnotsignificant.c.-40+*-Y-**-*-30+----*20+----*10+------+---------+---------+---------+---------+---------+X20.025.030.035.040.045.0Thescatterdiagramsuggeststhatacurvilinearrelationshipmaybeappropriate.d.TheMinitaboutputisshownbelow:TheregressionequationisY=-169+12.2X-0.177XSQPredictorCoefStdevt-ratiopConstant-168.8839.79-4.240.024X12.1872.6634.580.020XSQ-0.177040.04290-4.130.026s=3.248R-sq=94.5%R-sq(adj)=90.8%AnalysisofVarianceSOURCEDFSSMSFpRegression2541.85270.9225.680.013Error331.6510.55Total5573.5013-338 e.Sincethep-valuecorrespondingtoF=25.68is.013<therelationshipissignificant.2f.yˆ=-168.88+12.187(25)-0.17704(25)=25.1452.a.TheMinitaboutputisshownbelow:TheregressionequationisY=9.32+0.424XPredictorCoefStdevt-ratiopConstant9.3154.1962.220.113X0.42420.19442.180.117s=3.531R-sq=61.4%R-sq(adj)=48.5%AnalysisofVarianceSOURCEDFSSMSFpRegression159.3959.394.760.117Error337.4112.47Total496.80Thehighp-value(.117)indicatesaweakrelationship;notethat61.4%ofthevariabilityinyhasbeenexplainedbyx.b.TheMinitaboutputisshownbelow:TheregressionequationisY=-8.10+2.41X-0.0480XSQPredictorCoefStdevt-ratiopConstant-8.1014.104-1.970.187X2.41270.44095.470.032XSQ-0.047970.01050-4.570.045s=1.279R-sq=96.6%R-sq(adj)=93.2%AnalysisofVarianceSOURCEDFSSMSFpRegression293.52946.76528.600.034Error23.2711.635Total496.800Atthe.05levelofsignificance,therelationshipissignificant;thefitisexcellent.2c.yˆ=-8.101+2.4127(20)-0.04797(20)=20.9653.a.Thescatterdiagramshowssomeevidenceofapossiblelinearrelationship.b.TheMinitaboutputisshownbelow:TheregressionequationisY=2.32+0.637XPredictorCoefStdevt-ratiopConstant2.3221.8871.230.25813-339 X0.63660.30442.090.075s=2.054R-sq=38.5%R-sq(adj)=29.7%AnalysisofVarianceSOURCEDFSSMSFpRegression118.46118.4614.370.075Error729.5394.220Total848.000c.Thefollowingstandardizedresidualplotindicatesthattheconstantvarianceassumptionisnotsatisfied.--*-1.2+*---*-**0.0+--**---1.2+-**--+---------+---------+---------+---------+---------+------YHAT3.04.05.06.07.08.0d.Thelogarithmictransformationdoesnotappeartoeliminatethewedged-shapedpatternintheaboveresidualplot.Thereciprocaltransformationdoes,however,removethewedge-shapedpattern.Neithertransformationprovidesagoodfit.TheMinitaboutputforthereciprocaltransformationandthecorrespondingstandardizedresidualpotareshownbelow.Theregressionequationis1/Y=0.275-0.0152XPredictorCoefStdevt-ratiopConstant0.274980.046015.980.000X-0.0151820.007421-2.050.080s=0.05009R-sq=37.4%R-sq(adj)=28.5%AnalysisofVarianceSOURCEDFSSMSFpRegression10.0105010.0105014.190.080Error70.0175630.002509Total80.028064-*---1.0+*-*-13-340 --*0.0+*---**--1.0+-**---+---------+---------+---------+---------+---------+----YHAT0.1400.1600.1800.2000.2200.2404.a.TheMinitaboutputisshownbelow:TheregressionequationisY=943+8.71XPredictorCoefStdevt-ratiopConstant943.0559.3815.880.000X8.7141.5445.640.005s=32.29R-sq=88.8%R-sq(adj)=86.1%AnalysisofVarianceSOURCEDFSSMSFpRegression1332233322331.860.005Error441721043Total537395b.Thep-valueof.005<=.01;rejectH05.TheMinitaboutputisshownbelow:TheregressionequationisY=433+37.4X-0.3831/YPredictorCoefStdevt-ratiopConstant432.6141.23.060.055X37.4297.8074.790.0171/Y-0.38290.1036-3.700.034s=15.83R-sq=98.0%R-sq(adj)=96.7%AnalysisofVarianceSOURCEDFSSMSFpRegression2366431832273.150.003Error3751250Total53739513-341 b.Sincethelinearrelationshipwassignificant(Exercise4),thisrelationshipmustbesignificant.Notealsothatsincethep-valueof.005<=.05,wecanrejectH0.c.Thefittedvalueis1302.01,withastandarddeviationof9.93.The95%confidenceintervalis1270.41to1333.61;the95%predictionintervalis1242.55to1361.47.6.a.Thescatterdiagramisshownbelow:-*1.60+-DISTANCE---1.20+-*---0.80+*--*--**0.40++---------+---------+---------+---------+---------+------NUMBER8.012.016.020.024.028.0b.No;therelationshipappearstobecurvilinear.c.Severalpossiblemodelscanbefittedtothesedata,asshownbelow:22yˆ=2.90-0.185x+.00351xR.91a12yˆ0.046814.4R.91ax7.a.TheMinitaboutputisshownbelow:-36+x-Shipment---24+-x--x-x12+--xx-2xx-0++---------+---------+---------+---------+---------+-----Media$13-342 0255075100125b.TheMinitaboutputisshownbelow:TheregressionequationisShipment=4.09+0.196Media$PredictorCoefSECoefTPConstant4.0892.1681.890.096Media$0.195520.036355.380.000S=5.044R-Sq=78.3%R-Sq(adj)=75.6%AnalysisofVarianceSourceDFSSMSFPRegression1735.84735.8428.930.000Error8203.5125.44Total9939.35UnusualObservationsObsMedia$ShipmentFitStDevFitResidualStResid112036.3027.553.308.752.30RRdenotesanobservationwithalargestandardizedresidualSimplelinearregressionappearstodogoodjobinexplainingthevariabilityinshipments.However,thescatterdiagraminpart(a)indicatesthatacurvilinearrelationshipmaybemoreappropriate.c.TheMinitaboutputisshownbelow:TheregressionequationisShipment=5.51+0.00182Media$SqPredictorCoefSECoefTPConstant5.5061.6863.270.011Media$Sq0.00182250.00027926.530.000S=4.308R-Sq=84.2%R-Sq(adj)=82.2%AnalysisofVarianceSourceDFSSMSFPRegression1790.88790.8842.620.000Error8148.4718.56Total9939.35UnusualObservationsObsMedia$SqShipmentFitStDevFitResidualStResid31002015.9023.772.26-7.87-2.15RRdenotesanobservationwithalargestandardizedresidual13-343 8.a.Thescatterdiagramisshownbelow:8070605040302005ProjectedUsers(%)20100010203040501999InternetUsers(%)Itappearsthatasimplelinearregressionmodelisnotappropriatebecausethereiscurvatureintheplot.b.TheMinitaboutputisshownbelow:Theregressionequationis2005%=17.1+3.151999%-0.04451999%SqPredictorCoefSECoefTPConstant17.0994.6393.690.0031999%3.14620.49716.330.0001999%Sq-0.044540.01018-4.370.001S=5.667R-Sq=89.7%R-Sq(adj)=88.1%AnalysisofVarianceSourceDFSSMSFPRegression23646.31823.256.780.000ResidualError13417.432.1Total154063.8c.TheMinitaboutputisshownbelow:TheregressionequationisLog2000%=1.17+0.449Log1999%PredictorCoefSECoefTP13-344 Constant1.174200.0746815.720.000Log1999%0.448950.059787.510.000S=0.08011R-Sq=80.1%R-Sq(adj)=78.7%AnalysisofVarianceSourceDFSSMSFPRegression10.361990.3619956.400.000ResidualError140.089850.00642Total150.45184d.Theestimatedregressioninpart(b)ispreferredbecauseitexplainsahigherpercentageofthevariabilityinthedependentvariable.9.a.TheMinitaboutputisshownbelow:TheregressionequationisDistance=268+1.52Wind-0.0177WindSqPredictorCoefSECoefTPConstant267.8412.358113.600.000Wind1.521430.0771819.710.000WindSq-0.0176980.004456-3.970.001S=7.073R-Sq=95.7%R-Sq(adj)=95.3%AnalysisofVarianceSourceDFSSMSFPRegression22023310117202.200.000Error1890150Total20211342b.Theestimateddistanceis267.841+1.52143(-15)-0.017698(15)=241.04.2c.Theestimateddistanceis267.841+1.52143(25)-0.017698(25)=269.34.10.a.SSR=SST-SSE=1030MSR=1030MSE=520/25=20.8F=1030/20.8=49.52F.05=4.24(25DF)Since49.52>4.24werejectH0:andconcludethatx1issignificant.(520100)/2b.F48.3100/23F.05=3.42(2degreesoffreedomnumeratorand23denominator)Since48.3>3.42theadditionofvariablesx2andx3isstatisticallysignificant13-345 11.a.SSE=SST-SSR=1805-1760=45MSR=1760/4=440MSE=45/25=1.8F=440/1.8=244.44F.05=2.76(4degreesoffreedomnumeratorand25denominator)Since244.44>2.76,variablesx1andx4contributesignificantlytothemodelb.SSE(x1,x2,x3,x4)=45c.SSE(x2,x3)=1805-1705=100(10045)/2d.F15.281.8F.05=3.39(2numeratorand25denominatorDF)Since15.28>3.39weconcludethatx1andx3contributesignificantlytothemodel.12.a.TheMinitaboutputisshownbelow.TheregressionequationisPoints=170+6.61TeamIntPredictorCoefSECoefTPConstant170.1344.023.860.002TeamInt6.6132.2582.930.013S=43.93R-Sq=41.7%R-Sq(adj)=36.8%AnalysisofVarianceSourceDFSSMSFPRegression116546165468.570.013ResidualError12231571930Total1339703UnusualObservationsObsTeamIntPointsFitSEFitResidualStResid1333.0340.0388.434.2-48.4-1.75XXdenotesanobservationwhoseXvaluegivesitlargeinfluence.b.TheMinitaboutputisshownbelow.TheregressionequationisPoints=280+5.18TeamInt-0.0037Rushing-3.92OpponIntPredictorCoefSECoefTPConstant280.3481.423.440.006TeamInt5.1762.0732.500.032Rushing-0.003730.03336-0.110.913OpponInt-3.9181.651-2.370.039S=37.84R-Sq=63.9%R-Sq(adj)=53.1%13-346 AnalysisofVarianceSourceDFSSMSFPRegression32538684625.910.014ResidualError10143171432Total1339703SourceDFSeqSSTeamInt116546Rushing1776OpponInt18064SSE(reduced)-SSE(full)23,15714,317#extraterms2c.F3.09MSE(full)1432F.05=4.10(2numeratorand10denominatorDF)Since3.09<4.10theadditionofthetwoindependentvariablesisnotsignificant.Note:Supposethatweconsideredaddingonlythenumberofinterceptionsmadebytheopponents;thecorrespondingMinitaboutputisshownbelow:Points=274+5.23TeamInt-3.96OpponIntPredictorCoefSECoefTPConstant273.7753.815.090.000TeamInt5.2271.9312.710.020OpponInt-3.9651.524-2.600.025S=36.10R-Sq=63.9%R-Sq(adj)=57.3%AnalysisofVarianceSourceDFSSMSFPRegression225368126849.730.004ResidualError11143351303Total1339703SourceDFSeqSSTeamInt116546OpponInt18822UnusualObservationsObsTeamIntPointsFitSEFitResidualStResid1017.0312.00247.6416.8564.362.02RRdenotesanobservationwithalargestandardizedresidualInthiscase,23,15714,3351F6.771303F.05=4.84(1numeratorand11denominatorDF)13-347 Since6.77>4.84theadditionofthenumberofinterceptionsmadebytheopponentsissignificant.13.a.TheMinitaboutputisshownbelow:Points=218+0.0252Passing+4.39TeamInt-4.38OpponIntPredictorCoefSECoefTPConstant218.3869.073.160.010Passing0.025200.020391.240.245TeamInt4.3872.0052.190.053OpponInt-4.3761.525-2.870.017S=35.26R-Sq=68.7%R-Sq(adj)=59.3%AnalysisofVarianceSourceDFSSMSFPRegression32726990907.310.007ResidualError10124351243Total1339703SourceDFSeqSSPassing13416TeamInt113617OpponInt110235UnusualObservationsObsPassingPointsFitSEFitResidualStResid103247312.00247.8916.4664.112.06RRdenotesanobservationwithalargestandardizedresidualb.TheMinitaboutputisshownbelow:Points=235+0.0266Passing+4.18TeamInt-4.26OpponInt-0.0115RushingPredictorCoefSECoefTPConstant235.4087.522.690.025Passing0.026630.021741.220.252TeamInt4.1852.1801.920.087OpponInt-4.2561.635-2.600.029Rushing-0.011450.03316-0.350.738S=36.93R-Sq=69.1%R-Sq(adj)=55.4%AnalysisofVarianceSourceDFSSMSFPRegression42743168585.030.021ResidualError9122721364Total1339703SourceDFSeqSSPassing13416TeamInt113617OpponInt11023513-348 Rushing1163SSE(reduced)-SSE(full)12,43512,272#extraterms1c.F.1195MSE(full)1364F.05=5.12(1numeratorand9denominatorDF)Since.1195<5.12theadditionofRushingisnotsignificant.Note:Sinceonly1variablewasaddedtothemodelinpart(a),thetestcanalsobeperformedusingthet-ratioforRushingintheMinitaboutput.14.a.TheMinitaboutputisshownbelow:Risk=-111+1.32Age+0.296PressurePredictorCoefSECoefTPConstant-110.9416.47-6.740.000Age1.31500.17337.590.000Pressure0.296400.051075.800.000S=6.908R-Sq=80.6%R-Sq(adj)=78.4%AnalysisofVarianceSourceDFSSMSFPRegression23379.61689.835.410.000ResidualError17811.347.7Total194190.9SourceDFSeqSSAge11772.0Pressure11607.7UnusualObservationsObsAgeRiskFitSEFitResidualStResid1766.08.0025.051.67-17.05-2.54RRdenotesanobservationwithalargestandardizedresidualb.TheMinitaboutputisshownbelow:Risk=-123+1.51Age+0.448Pressure+8.87Smoker-0.00276AgePressPredictorCoefSECoefTPConstant-123.1656.94-2.160.047Age1.51300.77961.940.071Pressure0.44830.34571.300.214Smoker8.8663.0742.880.011AgePress-0.0027560.004807-0.570.575S=5.881R-Sq=87.6%R-Sq(adj)=84.3%AnalysisofVariance13-349 SourceDFSSMSFPRegression43672.11918.0326.540.000ResidualError15518.8434.59Total194190.95SourceDFSeqSSAge11771.98Pressure11607.66Smoker1281.10AgePress111.37UnusualObservationsObsAgeRiskFitSEFitResidualStResid1766.08.0020.912.01-12.91-2.34RRdenotesanobservationwithalargestandardizedresidualSSE(reduced)-SSE(full)811.3518.84#extraterms2c.F4.23MSE(full)34.59F.05=3.68(2numeratorand15denominatorDF)Since4.23>3.68theadditionofthetwotermsissignificant.15.a.LetPos1=0andPos2=0ifplayerisaguard;Pos1=1andPos2=0ifplayerisanoffensivetackle;Pos1=0andPos2=1ifplayerisawidereceiver.b.TheMinitaboutputisshownbelow:TheregressionequationisRating=12.5+0.706Pos1+1.92Pos2+0.0242Weight-2.63SpeedPredictorCoefSECoefTPConstant12.4904.2052.970.005Pos10.70550.29232.410.021Pos21.91790.92112.080.045Weight0.0241570.0076653.150.003Speed-2.63180.8600-3.060.004S=0.7111R-Sq=52.2%R-Sq(adj)=46.7%AnalysisofVarianceSourceDFSSMSFPRegression419.29994.82509.540.000ResidualError3517.69790.5057Total3936.9978c.Thep-valuecorrespondingtoF=9.54is0.000<=.05;thus,theestimatedregressionequationissignificant.d.TheMinitaboutputusingonlyWeightandSpeedisshownbelow:TheregressionequationisRating=19.8+0.0183Weight-3.59SpeedPredictorCoefSECoefTPConstant19.8392.8007.090.00013-350 Weight0.0182970.0060153.040.004Speed-3.59430.8576-4.190.000S=0.7763R-Sq=39.7%R-Sq(adj)=36.5%AnalysisofVarianceSourceDFSSMSFPRegression214.70167.350812.200.000ResidualError3722.29610.6026Total3936.9978TheFstatisticusedtodetermineiftheadditionofPos1andPos2resultsinasignificantreductionintheerrorsumofsquaresis22.296117.69792F4.55.5057TheFtablesintheappendixdonotshowavaluefortwonumeratorand35denominatordegreesoffreedom.But,theFvaluefortwonumeratorand30denominatordegreesoffreedomis3.32andtheFvaluefortwonumeratorand40denominatordegreesoffreedomis3.23.Thus,theFvaluefortwonumeratorand35denominatordegreesoffreedomisbetween3.23and3.32.BecausethecomputedFexceedsthisvalue,theadditionofPos1andPos2isstatisticallysignificant.Inotherwords,positionisasignificantfactorintheplayer’srating.16.a.TheMinitaboutputisshownbelow:Theregressionequationis%College=-26.6+0.0970SATScorePredictorCoefSECoefTPConstant-26.6137.22-0.720.485SATScore0.097030.037342.600.019S=12.83R-Sq=29.7%R-Sq(adj)=25.3%AnalysisofVarianceSourceDFSSMSFPRegression11110.81110.86.750.019ResidualError162632.3164.5Total173743.1UnusualObservationsObsSATScore%CollegeFitSEFitResidualStResid371640.0042.8610.79-2.86-0.41XXdenotesanobservationwhoseXvaluegivesitlargeinfluence.b.Alpha-to-Enter:0.15Alpha-to-Remove:0.15Responseis%Collegeon5predictors,withN=18Step1213-351 Constant-26.61-26.93SATScore0.0970.084T-Value2.602.46P-Value0.0190.026%TakeSAT0.204T-Value2.21P-Value0.043S12.811.5R-Sq29.6846.93R-Sq(adj)25.2839.86C-p6.93.8c.Backwardeliminationprocedure:Backwardelimination.Alpha-to-Remove:0.1Responseis%Collegeon5predictors,withN=18Step1234Constant33.7117.46-32.47-26.93Size-1.56-1.39T-Value-1.43-1.42P-Value0.1790.178Spending-0.0024-0.0026-0.0019T-Value-1.47-1.75-1.31P-Value0.1680.1040.212Salary-0.00026T-Value-0.40P-Value0.693SATScore0.0770.0810.0950.084T-Value2.062.362.772.46P-Value0.0620.0340.0150.026%TakeSAT0.2850.2740.2910.204T-Value2.472.532.602.21P-Value0.0290.0250.0210.043S11.210.911.211.5R-Sq59.6559.1052.7146.93R-Sq(adj)42.8346.5142.5839.86C-p6.04.24.13.8d.Responseis%CollegeSS%pAT13-352 eSTanaSkSdlceiiaoSznrrAVarsR-SqR-Sq(adj)C-pSegyeT129.725.36.912.826X125.520.88.213.203X246.939.93.811.508XX238.230.06.412.417XX352.742.64.111.244XXX349.538.75.011.618XXX459.146.54.210.852XXXX452.838.36.011.660XXXX559.642.86.011.219XXXXX17.a.Thecorrelationcoefficientsareasfollows:WinsPointsRushingPassingTeamIntPoints-0.6640.010Rushing0.527-0.3180.0530.268Passing0.2060.2930.1330.4790.3090.651TeamInt-0.6710.646-0.2850.2900.0090.0130.3240.314OpponInt0.506-0.6310.3120.120-0.2760.0650.0150.2780.6820.340CellContents:PearsoncorrelationP-ValueThevariablemosthighlycorrelatedwithWinsisTeamInt.TheMinitaboutputforthismodelusingTeamInttopredictWinsisshownbelow:TheregressionequationisWins=14.3-0.373TeamIntPredictorCoefSECoefTPConstant14.2942.3186.170.000TeamInt-0.37300.1189-3.140.009S=2.313R-Sq=45.1%R-Sq(adj)=40.5%AnalysisofVarianceSourceDFSSMSFPRegression152.65252.6529.840.009ResidualError1264.2055.350Total13116.857UnusualObservationsObsTeamIntWinsFitSEFitResidualStResid13-353 415.04.0008.6980.765-4.698-2.15R1333.04.0001.9831.8002.0171.39XRdenotesanobservationwithalargestandardizedresidualXdenotesanobservationwhoseXvaluegivesitlargeinfluence.b.Stepwiseregressionprocedure:Alpha-to-Enter:0.15Alpha-to-Remove:0.15ResponseisWinson5predictors,withN=14Step123Constant14.2947.58511.199TeamInt-0.37-0.44-0.28T-Value-3.14-4.14-2.45P-Value0.0090.0020.034Passing0.002560.00288T-Value2.273.02P-Value0.0440.013Points-0.026T-Value-2.37P-Value0.040S2.311.991.67R-Sq45.0662.6376.04R-Sq(adj)40.4855.8368.85C-p11.36.53.3c.Backwardeliminationprocedure:ResponseisWinson5predictors,withN=14Step123Constant8.0727.82711.199Points-0.024-0.023-0.026T-Value-1.54-2.08-2.37P-Value0.1630.0670.040Rushing0.00180.0018T-Value1.131.19P-Value0.2920.263Passing0.002610.002570.00288T-Value2.362.653.02P-Value0.0460.0260.01313-354 TeamInt-0.26-0.26-0.28T-Value-2.11-2.30-2.45P-Value0.0680.0470.034OpponInt-0.01T-Value-0.08P-Value0.939S1.741.641.67R-Sq79.3379.3176.04R-Sq(adj)66.4170.1268.85C-p6.04.03.3d.ResponseisWinsORPTpPuaepossaoihsmnniiIItnnnnVarsR-SqR-Sq(adj)C-pSsggtt145.140.511.32.3131X144.039.411.72.3344X262.655.86.51.9925XX261.654.76.82.0187XX376.068.93.31.6732XXX369.360.15.91.8932XXX479.370.14.01.6389XXXX476.065.45.31.7637XXXX579.366.46.01.7377XXXXX18.a.TheMinitaboutputisshownbelow:TheregressionequationisScoreAvg=81.4-0.147Green%PredictorCoefSECoefTPConstant81.4241.93042.200.000Green%-0.146810.02775-5.290.000S=0.4380R-Sq=60.9%R-Sq(adj)=58.7%AnalysisofVarianceSourceDFSSMSFPRegression15.37015.370127.990.000Error183.45370.1919Total198.8238UnusualObservationsObsGreen%ScoreAvgFitStDevFitResidualStResid378.670.350069.88490.27230.46511.36X1969.172.140071.27970.09840.86032.02R13-355 RdenotesanobservationwithalargestandardizedresidualXdenotesanobservationwhoseXvaluegivesitlargeinfluence.b.TheMinitaboutputisshownbelow:TheregressionequationisScoreAvg=58.2-0.00996Distance-0.152Green%+0.869PuttsPredictorCoefSECoefTPConstant58.1986.6448.760.000Distance-0.0099560.009111-1.090.291Green%-0.151860.02437-6.230.000Putts0.86860.22513.860.001S=0.3306R-Sq=80.2%R-Sq(adj)=76.5%AnalysisofVarianceSourceDFSSMSFPRegression37.07552.358521.580.000Error161.74830.1093Total198.8238UnusualObservationsObsDistanceScoreAvgFitStDevFitResidualStResid325670.350069.73690.21040.61312.40RRdenotesanobservationwithalargestandardizedresidualc.Theestimatedregressionequationappearstobereasonable;thatis,increasingDistanceandGreen%lowerstheaveragescore,whereasincreasingPuttsincreasestheaveragescore.d.Estimatedaveragescore=58.198-0.009956(231.6)-0.15186(65.2)+0.8686(30.69)=72.65.19.LetHealth=1ifhealth-drugsHealth=0ifenergy-internationalorotherTheregressionequationisP/E=10.8+0.430Sales%+10.6HealthPredictorCoefSECoefTPConstant10.8173.1433.440.004Sales%0.42970.18132.370.034Health10.6002.7503.850.002S=5.012R-Sq=55.4%R-Sq(adj)=48.5%AnalysisofVarianceSourceDFSSMSFPRegression2405.69202.858.070.005Error13326.5925.12Total15732.2813-356 SourceDFSeqSSSales%132.36Health1373.3320.SeethesolutiontoExercise14inthischapter.TheMinitaboutputusingthebestsubsetsregressionprocedureisshownbelow:ResponseisRiskPArgeSesmPsorAukegresVarsR-SqR-Sq(adj)C-pSeers163.361.328.59.2430X146.343.349.111.182X280.678.49.56.9083XX279.577.110.87.1058XX387.385.03.35.7566XXX386.283.74.76.0051XXX487.684.35.05.8813XXXXThisoutputsuggeststhatthemodelinvolvingAge,Pressure,andSmokeristhepreferredmodel;theMinitaboutputforthismodelisshownbelow:Risk=-91.8+1.08Age+0.252Pressure+8.74SmokerPredictorCoefSECoefTPConstant-91.7615.22-6.030.000Age1.07670.16606.490.000Pressure0.251810.045235.570.000Smoker8.7403.0012.910.010S=5.757R-Sq=87.3%R-Sq(adj)=85.0%AnalysisofVarianceSourceDFSSMSFPRegression33660.71220.236.820.000ResidualError16530.233.1Total194190.9SourceDFSeqSSAge11772.0Pressure11607.7Smoker1281.1UnusualObservationsObsAgeRiskFitSEFitResidualStResid1766.08.0021.111.94-13.11-2.42R13-357 21.a.TheMinitaboutputisshownbelow:TheregressionequationisP/E=6.51+0.569%PROFITPredictorCoefSECoefTpConstant6.5071.5094.310.000%PROFIT0.56910.12814.440.000s=2.580R-sq=53.7%R-sq(adj)=51.0%AnalysisofVarianceSOURCEDFSSMSFpRegression1131.40131.4019.740.000Error17113.146.66Total18244.54b.Theresidualplotasafunctionoftheorderinwhichthedataarepresentedisshownbelow:RESIDUAL---33.50+6-8-71-13-420.00+09-27-5-5-69-3.50+8--4-+---------+---------+---------+---------+05101520Theredoesnotappeartobeanypatternindicativeofpositiveautocorrelation.c.TheDurban-Watsonstatistic(obtainedfromMinitab)isd=2.34.Atthe.05levelofsignificance,dL=1.18anddU=1.39.Sinced>dUthereisnosignificantpositiveautocorrelation.22.FromMinitab,d=1.60.Atthe.05levelofsignificance,dL=1.04anddU=1.77.SincedLd,thetestisinconclusive.23.Thedummyvariablesaredefinedasfollows:x1x2x3Treatment000A100B010C001DE(y)=0+1x1+2x2+3x313-358 24.Thedummyvariablesaredefinedasfollows:x1x2Treatment001102013x3=0ifblock1and1ifblock2E(y)=0+1x1+2x2+3x325.FactorAx1=0iflevel1and1iflevel2FactorBx2x3Level001102013E(y)=0+1x1+2x2+3x1x2+4x1x326.a.Thedummyvariablesaredefinedasfollows:D1D2Mfg.001102013E(y)=0+1D1+2D2b.TheMinitaboutputisshownbelow:TheregressionequationisTIME=23.0+5.00D1-2.00D2PredictorCoefSECoefTpConstant23.0001.10620.800.000D15.0001.5633.200.011D2-2.0001.563-1.280.233s=2.211R-sq=70.3%R-sq(adj)=63.7%AnalysisofVarianceSOURCEDFSSMSFpRegression2104.00052.00010.640.004Error944.0004.889Total11148.000c.H0:1=2=0d.Thep-valueof.004islessthan=.05;therefore,wecanrejectH0andconcludethatthemeantimetomixabatchofmaterialismostthesameforeachmanufacturer.13-359 27.a.Thedummyvariablesaredefinedasfollows:D1D2D3Paint0001100201030014TheMinitaboutputisshownbelow:TheregressionequationisTIME=133+6.00D1+3.00D2+11.0D3PredictorCoefSECoefTpConstant133.0002.94145.220.000D16.0004.1591.440.168D23.0004.1590.720.481D311.0004.1592.640.018s=6.576R-sq=32.3%R-sq(adj)=19.6%AnalysisofVarianceSOURCEDFSSMSFpRegression3330.00110.002.540.093Error16692.0043.25Total191022.00Theappropriatehypothesistestis:H0:1=2=3=0Thep-valueof.093isgreaterthan=.05;therefore,atthe5%levelofsignificancewecannotrejectH0.b.Note:Estimatingthemeandryingforpaint2usingtheestimatedregressionequationsdevelopedinpart(a)maynotbethebestapproachbecauseatthe5%levelofsignificance,wecannotrejectH0.But,ifwewanttousetheoutput,wewouldprocedeasfollows.D1=1D2=0D3=0TIME=133+6(1)+3(0)+11(0)=13928.X1=0ifcomputerizedanalyzer,1ifelectronicanalyzerX2andX3aredefinedasfollows:X2X3Car001102013ThecompletedatasetandtheMinitaboutputareshownbelow:YX1X2X313-360 500005501063001421004411046101TheregressionequationisY=52.0-12.0X1+3.50X2+8.50X3PredictorCoefStdevt-ratiopConstant52.0002.64619.650.003X1-12.0002.646-4.540.045X23.5003.2401.080.393X38.5003.2402.620.120s=3.240R-sq=93.2%R-sq(adj)=83.1%AnalysisofVarianceSOURCEDFSSMSFpRegression3289.0096.339.170.100Error221.0010.50Total5310.00TotestforanysignificantdifferencebetweenthetwoanalyzerswemusttestH0:1Sincethep-valuecorrespondingtot=-4.54is.045<=.05,werejectH0:0thetimetodoatuneupisnotthesameforthetwoanalyzers.29.X1=0ifasmalladvertisementand1ifalargeadvertisementX2andX3aredefinedasfollows:X2X3Design00A10B01CThecompletedatasetandtheMinitaboutputareshownbelow:YX1X2X3X1X2X1X3800000120000013-361 121000081000022010001401000261101030110101000100180010018101011410101TheregressionequationisY=10.0+0.00X1+8.00X2+4.00X3+10.0X1X2+2.00X1X3PredictorCoefSECoefTpConstant10.0002.8283.540.012X10.0004.0000.001.000X28.0004.0002.000.092X34.0004.0001.000.356X1X210.0005.6571.770.128X1X32.0005.6570.350.736s=4.000R-sq=82.4%R-sq(adj)=67.6%AnalysisofVarianceSOURCEDFSSMSFpRegression5448.0089.605.600.029Error696.0016.00Total11544.0030.a.LetExSdenotetheinteractionbetweentheexpenseratio(%)andthesafetyrating.TheregressionequationisPerform%=23.3+222Expense%-28.9ExSPredictorCoefSECoefTPConstant23.3219.821.180.256Expense%222.4341.935.300.000ExS-28.8696.636-4.350.000S=13.01R-Sq=69.0%R-Sq(adj)=65.3%AnalysisofVarianceSourceDFSSMSFPRegression26396.43198.218.900.000ResidualError172876.6169.2ResidualError3517.69790.5057Total3936.9978Thetypeoffund(loadornoload)doesnotappeartobeasignificantfactorinpredictingtheone-yearperformance.Theinteractionbetweentheexpenseratioandthesafetyratingissignificant,andisaccountedforbytheExStermintheestimatedregressionequation.b.Thefitprovidedbytheestimatedregressionequationshowninpart(a)isnotbad;R-Sq(adj)=65.3%yˆ23.2222(1.12)28.9[(1.12)(7.6)]25.8;thus,theestimatedone-yearperformanceforAcornInternationalisapproximately26%.13-362 31.a.TheMinitaboutputisshownbelow:TheregressionequationisAUDELAY=80.4+11.9INDUS-4.82PUBLIC-2.62ICQUAL-4.07INTFINPredictorCoefStdevt-ratiopConstant80.4295.91613.600.000INDUS11.9443.7983.150.003PUBLIC-4.8164.229-1.140.263ICQUAL-2.6241.184-2.220.033INTFIN-4.0731.851-2.200.035s=10.92R-sq=38.3%R-sq(adj)=31.2%AnalysisofVarianceSOURCEDFSSMSFpRegression42587.7646.95.420.002Error354176.3119.3Total396764.0b.Thelowvalueoftheadjustedcoefficientofdetermination(31.2%)doesnotindicateagoodfit.c.Thescatterdiagramisshownbelow:96+-**AUDELAY-3-**-*80+*2*-**-3*-3*2-2*64+*-****-*-3-*48+*-2--+---------+---------+---------+---------+---------+----INTFIN0.01.02.03.04.05.0Thescatterdiagramsuggestsacurvilinearrelationshipbetweenthesetwovariables.d.Theoutputfromthestepwiseprocedureisshownbelow,whereINTFINSQisthesquareofINTFIN.ResponseisAUDELAYon5predictors,withN=40Step12Constant112.4112.8INDUS11.511.6T-Value3.673.80P-Value0.0010.001PUBLIC-1.0T-Value-0.2913-363 P-Value0.775ICQUAL-2.45-2.49T-Value-2.51-2.60P-Value0.0170.014INTFIN-36.0-36.6T-Value-4.61-4.91P-Value0.0000.000INTFINSQ6.56.6T-Value4.174.44P-Value0.0000.000S9.018.90R-Sq59.1559.05R-Sq(adj)53.1454.37C-p6.04.132.Thecomputeroutputisshownbelow:TheregressionequationisAUDELAY=63.0+11.1INDUSPredictorCoefSECoefTpConstant63.0003.39318.570.000INDUS11.0744.1302.680.011s=12.23R-sq=15.9%R-sq(adj)=13.7%AnalysisofVarianceSOURCEDFSSMSFpRegression11076.11076.17.190.011Error385687.9149.7Total396764.0UnusualObservationsObs.INDUSAUDELAYFitStdev.FitResidualSt.Resid50.0091.0063.003.3928.002.38R381.0046.0074.072.35-28.07-2.34RDurban-Watsonstatistic=1.55Atthe.05levelofsignificance,dL=1.44anddU=1.54.Sinced=1.55>dU,thereisnosignificantpositiveautocorrelation.33.a.TheMinitaboutputisshownbelow:TheregressionequationisAUDELAY=70.6+12.7INDUS-2.92ICQUALPredictorCoefSECoefTpConstant70.6344.55815.500.000INDUS12.7373.9663.210.003ICQUAL-2.9191.238-2.360.024s=11.56R-sq=26.9%R-sq(adj)=22.9%AnalysisofVariance13-364 SOURCEDFSSMSFpRegression21818.6909.36.800.003Error374945.4133.7Total396764.0SOURCEDFSEQSSINDUS11076.1ICQUAL1742.4UnusualObservationsObs.INDUSAUDELAYFitStdev.FitResidualSt.Resid50.0091.0067.713.7823.292.13R381.0046.0071.702.44-25.70-2.27RRdenotesanobs.withalargest.resid.Durban-Watsonstatistic=1.43b.Theresidualplotasafunctionoftheorderinwhichthedataarepresentedisshownbelow:RESID--5-617.5+0-8989-647-1-4670190.0+172480235-05-372-33-1946-17.5+2-5-8-+---------+---------+---------+---------+01020304013-365 Thereisnoobviouspatterninthedataindicativeofpositiveautocorrelation.c.Atthe.05levelofsignificance,dL=1.44anddU=1.54.Sinced=1.43>dU,thereisnosignificantpositiveautocorrelation.34.Thedummyvariablesaredefinedasfollows:D1D2Type00Non10Light01HeavyTheMinitaboutputisshownbelow:TheregressionequationisScore=4.25+1.00D1+1.50D2PredictorCoefSECoefTpConstant4.25000.381911.130.000D11.00000.54011.850.078D21.50000.54012.780.011s=1.080R-sq=27.6%R-sq(adj)=20.7%AnalysisofVarianceSOURCEDFSSMSFpRegression29.3334.6674.000.034Error2124.5001.167Total2333.833Sincethep-value=.034islessthan=.05,therearesignificantdifferencesbetweencomfortlevelsforthethreetypesofbrowsers.35.First,wewillusesimplelinearregressiontoestimatethechangeintheDowJonesIndustrialAverageusingjustparty.InthefollowingMinitaboutputthedummyvariablePartyiscodedasfollows:Party=0ifDemocrat,1ifRepublican.TheregressionequationisChangeDJ=7.13+0.92PartyPredictorCoefSECoefTPConstant7.1254.7681.490.146Party0.9206.0310.150.880S=16.52R-Sq=0.1%R-Sq(adj)=0.0%AnalysisofVarianceSourceDFSSMSFPRegression16.36.30.020.880Error308184.3272.8Total318190.6UnusualObservationsObsPartyChangeDJFitStDevFitResidualStResid101.00-27.608.053.69-35.65-2.21RRdenotesanobservationwithalargestandardizedresidual13-366 Therelationshipbetweenthesetwovariablesisnotstatisticallysignificant.Tomodeltheeffectoftheyearinthepresidentialterm,threedummyvariableswereused:Year1,Year2,andYear3.Thesevariableswerecodedasfollows:Year1=1iftheobservationcorrespondstothefirstyearinthetermofoffice,0otherwise;Year2=1iftheobservationcorrespondstothesecondyearinthetermofoffice,0otherwise;andYear3=1iftheobservationcorrespondstothethirdyearinthetermofoffice,0otherwise.UsingParty,Year1,Year2,andYear3asindependentvariables,theonlyvariablethatprovedtobesignificantatthe.05levelofsignificanceisYear3.TheMinitaboutputisshownbelow:TheregressionequationisChangeDJ=4.42+13.1Year3PredictorCoefSECoefTPConstant4.4253.1541.400.171Year313.1006.3072.080.046S=15.45R-Sq=12.6%R-Sq(adj)=9.7%AnalysisofVarianceSourceDFSSMSFPRegression11029.71029.74.310.046Error307161.0238.7Total318190.6UnusualObservationsObsYear3ChangeDJFitStDevFitResidualStResid100.00-27.604.423.15-32.02-2.12RRdenotesanobservationwithalargestandardizedresidualChapter17IndexNumbersLearningObjectives1.Knowhowtocomputepricerelativesandunderstandhowtheyrepresentpricechangesovertime.2.Knowhowtocomputeaggregatepriceindexesandunderstandhowthechoiceofabaseperiodaffectstheindex.3.BecomefamiliarwiththeConsumerPriceIndex,theProducerPriceIndexandtheDowJonesaverages.4.Learnhowtodeflateatimeseriestomeasurechangesovertimeinconstantdollars.5.Learnhowtocomputeanaggregatequantityindexandhowtointerpretit.13-367 Solutions:1.a.ItemPriceRelativeA103=(7.75/7.50)B238=(1500/630)7.751500.001507.75b.I(100)(100)23720017.50630.00637.507.75(1500)1500.00(2)14,625.00c.I(100)(100)11720017.50(1500)630.00(2)12,510.007.75(1800)1500.00(1)15,450.00d.I(100)(100)10920017.50(1800)630.00(1)14,130.002.a.Fromthepricerelativeweseethepercentageincreasewas32%.b.Dividethecurrentcostbythepricerelativeandmultiplyby100.$10.751990cost=(100)=$8.141323.a.PriceRelativesA=(6.00/5.45)100=11013-368 B=(5.95/5.60)100=106C=(6.20/5.50)100=1136.005.956.20b.I(100)11020015.455.605.506.00(150)5.95(200)6.20(120)c.I(100)10920015.45(150)5.60(200)5.50(120)9%increaseoverthetwoyearperiod.16.25(35,000)64.00(5,000)10.00(60,000)4.I(100)105200115.00(35,000)60.00(5,000)9.80(60,000).19(500)1.80(50)4.20(100)13.20(40)5.I(100)104.15(500)1.60(50)4.50(100)12.00(40)Paascheindex6.PriceBasePeriodBasePeriodWeightedPriceItemRelativePriceUsageWeightRelativesA15022.002044066,000B905.005025022,500C12014.004056067,2001250155,700155,700I12512507.a.PriceRelativesA=(3.95/2.50)100=158B=(9.90/8.75)100=113C=(.95/.99)100=96b.PriceBaseWeightWeightedPriceItemRelativesPriceQuantityPi0QiRelativesA1582.502562.59875B1138.7515131.314837C96.996059.45702253.23041413-369 30414I120253.2Costofrawmaterialsisup20%forthechemical.8.PriceBaseWeightedPriceStockRelativesPriceQuantityWeightRelativesHoliday11015.505007750852500NYElectric10918.502003700403300KYGas9726.75500133751297375PQSoaps10842.253001267513689003750039220753922075I10537500Portfolioup5%9.PriceBaseWeightedPriceItemRelativesPriceQuantityWeightRelativesBeer10815.0035,000525,00056,700,000Wine10760.005,000300,00032,100,000SoftDrink1029.8060,000588,00059,976,0001,413,000148,776,000148,776,000I1051,413,000$7.2710.a.Deflated1980wages:(100)$8.8282.4$14.36Deflated2000wages:(100)$8.32172.614.36b.(100)197.5Thepercentageincreaseinactualwagesis97.5%.7.278.32c.(100)94.3Thechangeinreadwagesisadecreaseof5.7%.8.8211.199611.76(100/156.9)=7.49199712.23(100/160.5)=7.62199812.84(100/163.0)=7.88199913.35(100/166.6)=8.01200013.82(100/172.6)=8.018.011.02theincreaseinrealwagesandsalariesfrom1998to2000is2%.7.8812.a.19973929(100/160.5)=244813-370 19984052(100/163.0)=248619994260(100/166.6)=2557Manufacturers"shipmentsareincreasingslightlyinconstantdollarswhendeflatedusingtheCPI.b.19973929(100/131.8)=298119984052(100/130.7)=310019994260(100/133.0)=3203c.ThePPIisabetterdeflatorsincemanufacturingshipmentsreflectpricespaidbymanufacturers.13.DeflatedYearRetailSales($)CPIRetailSales($)1982380,00096.5393,7821987520,000113.6457,7461992700,000140.3498,9311997870,000160.5542,0562000940,000172.6544,612Intermsofconstantdollars,thefirm"ssalesareincreasingmoderately.300(18.00)400(4.90)850(15.00)20,11014.I(100)(100)110350(18.00)220(4.90)730(15.00)18,32895(1200)75(1800)50(2000)70(1500)15.I(100)99120(1200)86(1800)35(2000)60(1500)Quantitiesaredownslightly.16.QuantityBaseWeightedQuantityModelRelativeQuantityPrice($)WeightRelativessSedan8520015,2003,040,000258,400,000Sport8010017,0001,700,000136,000,000Wagon807516,8001,260,000100,800,0006,000,000495,200,000495,200,000I836,000,00017.a/b.PriceIndexYear1996Base1997Base1996100.095.91997104.3100.01998108.9104.51999114.3109.618.a.PriceRelativesA=(15.90/10.50)(100)=15113-371 B=(32.00/16.25)(100)=197C=(17.40/12.20)(100)=143D=(35.50/20.00)(100)=17815.90(2000)32.00(5000)17.40(6500)35.50(2500)b.I(100)17010.50(2000)16.25(5000)12.20(6500)20.00(2500)15.90(4000)32.00(3000)17.40(7500)35.50(3000)19.I(100)16410.50(4000)16.25(3000)12.20(7500)20.00(3000)32.75(100)59(150)42(75)16.5(50)20.I(100)96Jan31.50(100)65(150)40(75)18(50)32.50(100)57.5(150)39.5(75)13.75(50)I(100)92Mar31.50(100)65(150)40(75)18(50)Marketisdowncomparedto1999.21.PriceRelatives:JanMarOil(32.75/31.50)(100)=104103Computer(59.00/65.00)(100)=9188Steel(42.00/40.00)(100)=10599RealEstate(16.5/18.00)(100)=9276IJan=96IMar=9222.BaseWeightedPriceProductRelativesPriceQuantityWeightRelativesCorn1132.3014273282370,866Soybeans1235.513501929237,2675211608,133608,133I117521123.a.FruitPriceRelativesBananas(.51/.41)(100)=124.4Apples(.85/.71)(100)=119.7Oranges(.61/.56)(100)=108.9Pears(.98/.64)(100)=153.1b.Weights(PioQio)PriceRelativeProduct9.963124.41239.397213-372 14.129119.71691.24137.784108.9847.67762.048153.1313.5488Totals33.9244091.86494091.8649I120.633.924Fruitpriceshaveincreasedby20.6%overthe10-yearperiodaccordingtotheindex.24.Salariesinconstant(1982-84)dollarsarecomputedasfollows:1970$14,000(100/38.8)=$36,0821975$17,500(100/53.8)=$32,5281980$23,000(100/82.4)=$27,9131985$37,000(100/107.6)=$34,3871990$53,000(100/130.7)=$40,5511995$65,000(100/152.4)=$42,6512000$80,000(100/172.6)=$46,350Inconstantdollarterms,realstartingsalarieshaveincreasedabout28%overthisperiod.25.Thestockmarketpricesinconstant(1982-84)dollarsarecomputedasfollows:1996$51.00(100/156.9)=$32.501997$54.00(100/160.5)=$33.641998$58.00(100/163.0)=$35.581999$59.50(100/166.6)=$35.712000$59.00(100/172.6)=$34.18Thevalueofthestock,inrealdollars,isonlyslightlymorein2000thanitisin1996.Ofcourse,ifthestockpaidahighdividenditmaystillhavebeenagoodinvestmentoverthisperiod.1200(30)500(20)500(25)26.I(100)143800(30)600(20)200(25)Quantityisup43%.Chapter18ForecastingLearningObjectives1.Understandthatthelong-runsuccessofanorganizationisoftencloselyrelatedtohowwellmanagementisabletopredictfutureaspectsoftheoperation.2.Knowthevariouscomponentsofatimeseries.13-373 3.Beabletousesmoothingtechniquessuchasmovingaveragesandexponentialsmoothing.4.Beabletousetheleastsquaresmethodtoidentifythetrendcomponentofatimeseries.5.Understandhowtheclassicaltimeseriesmodelcanbeusedtoexplainthepatternorbehaviorofthedatainatimeseriesandtodevelopaforecastforthetimeseries.6.Beabletodetermineanduseseasonalindexesforatimeseries.7.Knowhowregressionmodelscanbeusedinforecasting.8.Knowthedefinitionofthefollowingterms:timeseriesmeansquarederrorforecastmovingaveragestrendcomponentweightedmovingaveragescyclicalcomponentsmoothingconstantseasonalcomponentseasonalconstantirregularcomponentSolutions:1.a.Time-SeriesForecast2WeekValueForecastError(Error)182133154171252551615116916-7497513-374 Forecastforweek7is(17+16+9)/3=14b.MSE=75/3=25c.Smoothingconstant=.3.Time-SeriesValueForecastErrorSquaredError2WeektYtForecastFtYt-Ft(Yt-Ft)182138.005.0025.003159.006.0036.0041710.206.8046.2451611.564.4419.716912.45-3.4511.90138.85Forecastforweek7is.2(9)+.8(12.45)=11.76d.Forthe=.2exponentialsmoothingforecastMSE=138.85/5=27.77.Sincethethree-weekmovingaveragehasasmallerMSE,itappearstoprovidethebetterforecasts.e.Smoothingconstant=.4.Time-SeriesValueForecastErrorSquaredError2WeektYtForecastFtYt-Ft(Yt-Ft)182138.05.025.0031510.05.025.0041712.05.025.0051614.02.04.006914.8-5.833.64112.64MSE=112.64/5=22.53.Asmoothingconstantof.4appearstoprovidebetterforecasts.Forecastforweek7is.4(9)+.6(14.8)=12.482.a.4-WeekMoving5-WeekMovingTime-SeriesAverageAverage22WeekValueForecast(Error)Forecast(Error)11722131942351820.004.0061620.2518.0619.6012.9672019.001.0019.400.3613-375 81819.251.5619.201.4492218.0016.0019.009.00102019.001.0018.801.44111520.0025.0019.2017.64122218.7510.5619.009.0077.1851.84b.MSE(4-Week)=77.18/8=9.65MSE(5-Week)=51.84/7=7.41c.Forthelimiteddataprovided,the5-weekmovingaverageprovidesthesmallestMSE.3.a.Time-SeriesWeightedMovingForecast2WeekValueAverageForecastError(Error)11722131942319.333.6713.4751821.33-3.3311.0961619.83-3.8314.6772017.832.174.7181818.33-0.330.1192218.333.6713.47102020.33-0.330.11111520.33-5.3328.41122217.834.1717.39103.43b.MSE=103.43/9=11.49Prefertheunweightedmovingaveragehere.c.Youcouldalwaysfindaweightedmovingaverageatleastasgoodastheunweightedone.Actuallytheunweightedmovingaverageisaspecialcaseoftheweightedoneswheretheweightsareequal.4.2WeekTime-SeriesValueForecastError(Error)11722117.004.0016.0031917.401.602.5642317.565.4429.5951818.10-0.100.0161618.09-2.094.3772017.882.124.4981818.10-0.100.0192218.093.9115.2913-376 102018.481.522.31111518.63-3.6313.18122218.273.7313.91101.72MSE=101.72/11=9.25=.2providedalowerMSE;therefore=.2isbetterthan5.a.F13=.2Y12+.16Y11+.64(.2Y10+.8F10)=.2Y12+.16Y11+.128Y10+.512F10F13=.2Y12+.16Y11+.128Y10+.512(.2Y9+.8F9)=.2Y12+.16Y11+.128Y10+.1024Y9+.4096F9F13=.2Y12+.16Y11+.128Y10+.1024Y9+.4096(.2Y8+.8F8)=.2Y12+.16Y11+.128Y10+.1024Y9+.08192Y8+.32768F8b.Themorerecentdatareceivesthegreaterweightorimportanceindeterminingtheforecast.Themovingaveragesmethodweightsthelastndatavaluesequallyindeterminingtheforecast.6.a.3-MonthMoving=222MonthYtAveragesForecast(Error)Forecast(Error)18028280.004.0038480.4012.9648382.001.0081.123.5358383.000.0081.502.2568483.330.4581.804.8478583.332.7982.247.6288484.000.0082.791.4698284.335.4383.031.06108383.670.4582.830.03118483.001.0082.861.30128383.000.0083.090.0111.1239.06MSE(3-Month)=11.12/9=1.24MSE(=.2)=39.06/11=3.55Use3-monthmovingaverages.b.(83+84+83)/3=83.37.a.Time-Series3-MonthMoving4-MonthMoving22MonthValueAverageForecast(Error)AverageForecast(Error)19.529.339.449.69.400.0459.89.430.149.450.1269.79.600.019.530.0379.89.700.019.630.0313-377 810.59.770.539.730.5999.910.000.019.950.00109.710.070.149.980.08119.610.030.189.970.14129.69.730.029.920.101.081.09MSE(3-Month)=1.08/9=.12MSE(4-Month)=1.09/8=.14Use3-Monthmovingaverages.b.Forecast=(9.7+9.6+9.6)/3=9.63c.Forthelimiteddataprovided,the5-weekmovingaverageprovidesthesmallestMSE.8.a.Time-Series3-MonthMoving=.222MonthValueAverageForecast(Error)Forecast(Error)12402350240.0012100.003230262.001024.004260273.33177.69255.6019.365280280.000.00256.48553.196320256.674010.69261.183459.797220286.674444.89272.952803.708310273.331344.69262.362269.579240283.331877.49271.891016.9710310256.672844.09265.511979.3611240286.672178.09274.411184.0512230263.331110.89267.531408.5017,988.5227,818.49MSE(3-Month)=17,988.52/9=1998.72MSE(=.2)=27,818.49/11=2528.95BasedontheaboveMSEvalues,the3-monthmovingaveragesappearsbetter.However,exponentialsmoothingwaspenalizedbyincludingmonth2whichwasdifficultforanymethodtoforecast.Usingonlytheerrorsformonths4to12,theMSEforexponentialsmoothingisrevisedtoMSE(=.2)=14,694.49/9=1632.72Thus,exponentialsmoothingwasbetterconsideringmonths4to12.b.Usingexponentialsmoothing,F13=Y12+(1-)F12=.20(230)+.80(267.53)=2609.a.Smoothingconstant=.3.Time-SeriesValueForecastErrorSquaredError2MonthtYtForecastFtYt-Ft(Yt-Ft)1105.013-378 2135.0105.0030.00900.003120.0114.006.0036.004105.0115.80-10.80116.64590.0112.56-22.56508.956120.0105.7914.21201.927145.0110.0534.951221.508140.0120.5419.46378.699100.0126.38-26.38695.901080.0118.46-38.461479.1711100.0106.92-6.9247.8912110.0104.855.1526.52Total5613.18MSE=5613.18/11=510.29Forecastformonth13:F13=.3(110)+.7(104.85)=106.4b.Smoothingconstant=.5Time-SeriesValueForecastError2MonthtYtForecastFtYt-FtSquaredError(Yt-Ft)1105213510530.00900.003120.5(135)+.5(105)=1200.000.004105.5(120)+.5(120)=120-15.00225.00590.5(105)+.5(120)=112.50-22.50506.256120.5(90)+.5(112.5)=101.2518.75351.567145.5(120)+.5(101.25)=110.6334.371181.308140.5(145)+.5(110.63)=127.8112.19148.609100.5(140)+.5(127.81)=133.91-33.911149.891080.5(100)+.5(133.91)=116.95-36.951365.3011100.5(80)+.5(116.95)=98.481.522.3112110.5(100)+.5(98.48)=99.2410.76115.785945.99MSE=5945.99/11=540.55Forecastformonth13:F13=.5(110)+.5(99.24)=104.62Conclusion:asmoothingconstantof.3isbetterthanasmoothingconstantof.5sincetheMSEislessfor0.3.10.a/b.Time-Series=.2=.322WeekValueForecast(Error)Forecast(Error)17.3527.407.35.00257.35.002537.557.36.03617.36.036147.567.40.02567.42.019613-379 57.607.43.02897.46.019667.527.46.00367.50.000477.527.48.00167.51.000187.707.48.04847.51.036197.627.53.00817.57.0025107.557.55.00007.58.0009.1548.1178c.MSE(=.2)=.1548/9=.0172MSE(=.3)=.1178/9=.0131Use=.3.F11=.3Y10+.7F10=.3(7.55)+.7(7.58)=7.5711.a.MethodForecastMSE3-Quarter80.732.534-Quarter80.552.81The3-quartermovingaverageforecastisbetterbecauseithasthesmallestMSE.b.MethodForecastMSE=.480.402.40=.580.572.01The=.5smoothingconstantisbetterbecauseithasthesmallestMSE.c.The=.5isbetterbecauseithasthesmallestMSE.12.Thefollowingvaluesareneededtocomputetheslopeandintercept:2ttYt151555Y186tttY()tY/n186(15)(55)/5ttb2.1122tt2/n55(15)/5bYbt112.1(3)4.701Tt=4.7+2.1tForecast:T6=4.7+2.1(6)=17.313.Thefollowingvaluesareneededtocomputetheslopeandintercept:2ttYt21911171Y4037ttComputationofslope:13-380 tY()tY/n4037(21)(1171)/6ttb3.5143122tt2/n91(21)/6Computationofintercept:bYbt195.1667(3.5143)(3.5)207.466801Equationforlineartrend:Tt=207.467-3.514tForecast:T6=207.467-3.514(7)=182.8714.Thefollowingvaluesareneededtocomputetheslopeandintercept:2ttY2191117.1tY403.7ttComputationofslope:tY()tY/n403.7(21)(117.1)/6ttb0.3514122tt2/n91(21)/6Computationofintercept:bYbt19.5167(0.3514)(3.5)20.746601Equationforlineartrend:Tt=20.7466-0.3514tConclusion:enrollmentappearstobedecreasingbyanaverageofapproximately351studentsperyear.15.Thefollowingvaluesareneededtocomputetheslopeandintercept:2ttY28140213,400tY865,400ttComputationofslope:tY()tY/n865,400(28)(213,400)/7ttb421.429122tt2/n140(28)/7Computationofintercept:bYbt30,485.714421.429(4)28,80001Equationforlineartrend:Tt=28,800+421.429t16.Alineartrendmodelisnotappropriate.Anonlinearmodelwouldprovideabetterapproximation.17.a.Alineartrendappearstobereasonable.b.Thefollowingvaluesareneededtocomputetheslopeandintercept:13-381 2ttY36204223.8tY1081.6ttComputationofslope:tY()tY/n1081.6(36)(223.8)/8ttb1.7738122tt2/n204(36)/8Computationofintercept:bYbt27.9751.7738(4.5)19.99301Equationforlineartrend:Tt=19.993+1.774tConclusion:Thefirmhasbeenrealizinganaveragecostincreaseof$1.77perunitperyear.18.a.Thefollowingvaluesareneededtocomputetheslopeandintercept:2ttY5538514.26tY94.34ttComputationofslope:tY()tY/n94.35(55)(14.26)/10ttb.19297122tt2/n385(55)/10Computationofintercept:bYbt1.426.19297(5.5).36501Equationforlineartrend:Tt=.365+.193tForecast:Tt=.365+.193(11)=$2.49b.Overthepasttenyearstheearningspersharehavebeenincreasingattheaveragerateof$.193peryear.AlthoughthisisapositiveindicatorofWalgreen’sperformance.Moreinformationwouldbenecessarytoconclude“goodinvestment.”19.a.Thefollowingvaluesareneededtocomputetheslopeandintercept:2ttYt219145.5Y160.15ttComputationofslope:tY()tY/n160.15(21)(45.5)/6ttb0.0514122tt2/n91(21)/6Computationofintercept:bYbt7.5833-0.0514(3.5)=7.403301Equationforlineartrend:Tt=7.4033+0.0514t13-382 Thenumberofapplicationsisincreasingbyapproximately1630peryear.b.1996:Tt=7.4033+0.0514(7)=7.7633orabout7.76%1997:Tt=7.4033+0.0514(8)=7.8148orabout7.81%20.a.Thefollowingvaluesareneededtocomputetheslopeandintercept:2ttY5538541841tY262,923ttComputationofslope:tY()tY/n262,923(55)(41,841)/10ttb397.545122tt2/n385(55)/10Computationofintercept:bYbt4184.1-397.545(5.5)=1997.601Equationforlineartrend:Tt=1997.6+397.545tb.T11=1997.6+397.545(11)=6371T12=1997.6+397.545(12)=676821.a.Thefollowingvaluesareneededtocomputetheslopeandintercept:2ttY2191118.2tY549.7ttComputationofslope:tY()tY/n549.7(21)(118.2)/6ttb7.7714122tt2/n91(21)/6Computationofintercept:bYbt(118.2/6)-7.7714(21/6)=-7.501Equationforlineartrend:Tt=-7.5+7.7714tb.7.7714($M)peryearc.1998forecast:T8=-7.5+7.7714(7)=46.922.a.Four-QuarterCenteredYearQuarterYtMovingAverageMovingAverage11413-383 223.50333.7504.00454.1254.252164.5004.75235.0005.25355.3755.50475.8756.253176.3756.50266.6256.753648b.CenteredSeasonal-IrregularYearQuarterYtMovingAverageComponent11422333.7500.8000454.1251.21212164.5001.3333235.0000.6000355.3750.9302475.8751.19153176.3751.0980266.6250.90573648Seasonal-IrregularAdjustedSeasonalQuarterComponentValuesSeasonalIndexIndex13-384 11.3333,1.09801.21571.20502.60000,.90570.75290.74633.80000,.90320.86510.867541.2121,1.19151.20181.19124.0355Note:Adjustmentforseasonalindex=4.000/4.0355=0.991223.a.Fourquartermovingaveragesbeginningwith(1690+940+2625+2500)/4=1938.75Othermovingaveragesare1966.252002.501956.252052.502025.002060.001990.002123.75b.Seasonal-IrregularAdjustedSeasonalQuarterComponentValuesSeasonalIndexIndex10.9040.9000.90200.90020.4480.5260.49700.48631.3441.4531.39851.39641.2751.1641.21951.2174.0070Note:Adjustmentforseasonalindex=4.000/4.007=0.9983c.Thelargestseasonaleffectisinthethirdquarterwhichcorrespondstotheback-to-schooldemandduringJuly,August,andSeptemberofeachyear.24.Seasonal-IrregularMonthComponentValuesSeasonalIndexAdjustedSeasonalIndex10.720.700.710.70720.800.750.780.77730.830.820.830.82740.940.990.970.96651.011.021.021.01661.251.361.311.30571.491.511.501.49481.191.261.231.22590.980.970.980.976100.981.000.990.986110.930.940.940.936120.780.800.790.78712.05Notes:1.Adjustmentforseasonalindex=12/12.05=0.99613-385 2.Theadjustmentisreallynotnecessaryinthisproblemsinceitimpliesmoreaccuracythaniswarranted.Thatis,theseasonalcomponentvaluesandtheseasonalindexwereroundedtotwodecimalplaces.25.a.Useatwelveperiodmovingaverages.Aftercenteringthemovingaverages,youshouldobtainthefollowingseasonalindexes:HourSeasonalIndexHourSeasonalIndex10.77171.20720.86480.99430.95490.85041.392100.64751.571110.57961.667120.504b.ThehoursofJuly18arenumber37to48inthetimeseries.Thusthetrendcomponentfor7:00a.m.onJuly18(period37)wouldbeT37=32.983+.3922(37)=47.49AsummaryofthetrendcomponentsforthetwelvehoursonJuly18isasfollows:HourTrendComponentHourTrendComponent147.49749.85247.89850.24348.28950.63448.671051.02549.061151.42649.461251.81c.MultiplythetrendcomponentinpartbbytheseasonalindexesinpartatoobtainthetwelvehourlyforecastsforJuly18.Forexample,47.49x(.771)=36.6orroundedto37,wouldbetheforecastfor7:00a.m.onJuly18th.TheseasonallyadjustedhourlyforecastsforJuly18areasfollows:HourForecastHourForecast13776024185034694346810335771130682122626.a.Yes,thereisaseasonaleffectoverthe24hourperiod.TimePeriodSeasonalIndex12-4a.m.1.6964-8a.m.1.4588-120.71112-4p.m.0.3264-8p.m.0.4488-121.362b.13-386 TimePeriodForecast12-4p.m.166,761.134-8p.m.146,052.9927.a.3-MonthMovingForecast2MonthTime-SeriesValueAverageForecastError(Error)134.8750235.6250334.6875433.562535.0625-1.5002.2500532.625034.6250-2.0004.0000634.000033.62500.37500.1406733.625033.39580.22920.0525835.062533.41671.64582.7088934.062534.2292-0.16670.02781034.125034.2500-0.12500.01561133.250034.4167-1.16671.36111232.062533.8125-1.75003.0625Note:MSE=13.6189/9=1.51ForecastforDecemberis(34.1250+33.2500+32.0625)/3=33.1458b.Theweightedmovingaverageforecastsformonths4-12are35.1000,34.4250,33.4125,33.3625,33.5750,34.2750,34.3750,34.2875and33.7625.Note:MSE=12.3047/9=1.37ForecastforDecemberis0.2(34.125)+0.4(33.2500)+0.4(32.0625)=32.9500c.Theexponentialsmoothingforecastsformonths2-12are34.8750,35.1375,34.9800,34.4839,33.8333,33.8916,33.7983,34.2408,34.1784,34.1597and33.8413.Note:MSE=11.1881=1.24ForecastforDecemberis0.35(32.0625)+0.65(33.8413)=33.2187d.MethodMSEMovingAverage1.51WeightedMovingAverage1.37ExponentialSmoothing1.24ExponentialSmoothingisthebestofthethreeapproachesbecauseithasthesmallestMSE.13-387 28.a.Time-Series=0.1=0.1=0.2=0.222YearValueForecast(Error)Forecast(Error)19725519743855.0289.055.0289.019765453.30.551.65.819783753.4268.052.1227.419805351.71.649.115.519824051.9140.749.997.019845350.75.447.926.219863650.9222.248.9166.519885049.40.346.313.519903749.5155.647.1101.219925548.245.945.099.119943948.998.147.064.619964947.91.245.412.7Totals:1228.41118.5SmoothingConstantMSE0.11228.4/12=102.40.21118.5/12=93.2Asmoothingconstantof0.2isbetter.b.Using=0.2Forecastfor1998=0.2(49)+0.8(45.4)=46.129.a.TimeSeries=.2=.3=.4PeriodValueForecastsForecastsForecasts128.9231.029.8029.8029.80329.930.0430.1630.28430.130.0130.0830.13532.230.0330.0930.12631.530.4630.7230.95732.030.6730.9531.17831.930.9431.2731.50930.031.1331.4631.66MSE(=.2)=1.40MSE(=.3)=1.27MSE(=.4)=1.23=.4providesthebestforecastb.Using=.4,F10=.4(.30)+.6(31.66)=31.0013-388 30.Time-SeriesValueForecastForecastErrorSquaredError2WeektYtFtYt-Ft(Yt-Ft)12221822.00-4.0016.0032321.201.803.2442121.56-0.560.3151721.45-4.4519.8062420.563.4411.8372021.25-1.251.5681921.00-2.004.0091820.60-2.606.76102120.080.920.85Total64.35MSE=64.35/9=7.15Forecastforweek11:F11=0.2(21)+0.8(20.08)=20.2631.2tYtFtYt-Ft(Yt-Ft)12,75023,1002,750.00350.00122,500.0033,2502,890.00360.00129,600.0042,8003,034.00-234.0054,756.0052,9002,940.40-40.401,632.1663,0502,924.24125.7615,815.5873,3002,974.54325.46105,924.2183,1003,104.73-4.7322.3792,9503,102.84-152.8423,260.07103,0003,041.70-41.701,738.89113,2003,025.02174.9830,618.00123,1503,095.0154.993,023.90Total:488,991.18MSE=488,991.18/11=44,453.74Forecastforweek13:F13=0.4(3,150)+0.6(3,095.01)=3,117.0132.a.SmoothingConstantMSE=.34,492.37=.42,964.67=.52,160.31The=.5smoothingconstantisbetterbecauseithasthesmallestMSE.b.Tt=244.778+22.088tMSE=357.8113-389 c.TrendprojectionprovidesmuchbetterforecastsbecauseithasthesmallestMSE.ThereasonMSEissmallerfortrendprojectionisthatsalesareincreasingovertime;asaresult,exponentialsmoothingcontinuouslyunderestimatesthevalueofsales.Ifyoulookattheforecasterrorsforexponentialsmoothingyouwillseethattheforecasterrorsarepositiveforperiods2through18.33.a.ForecastforJulyis236.97ForecastforAugust,usingforecastforJulyastheactualsalesinJuly,is236.97.Exponentialsmoothingprovidesthesameforecastforeveryperiodinthefuture.Thisiswhyitisnotusuallyrecommendedforlong-termforecasting.b.Tt=149.719+18.451tForecastforJulyis278.88ForecastforAugustis297.33c.Theproposedsettlementisnotfairsinceitdoesnotaccountfortheupwardtrendinsales.Basedupontrendprojection,thesettlementshouldbebasedonforecastedlostsalesof$278,880inJulyand$297,330inAugust.34.Thefollowingvaluesareneededtocomputetheslopeandintercept:2ttYt281401575Y6491ttComputationofslope:tY()tY/n6491(28)(1575)/7ttb6.8214122tt2/n140(28)/7Computationofintercept:bYbt225-6.8214(4)=197.71401Equationforlineartrend:Tt=197.714+6.821tForecast:T8=197.714+6.821(8)=252.28T9=65.025+4.735(9)=259.1035.Thefollowingvaluesareneededtocomputetheslopeandintercept:2ttY1201240578,400tY5,495,900ttComputationofslope:tY()tY/n5,495,900(120)(578,400)/15ttb3102.5122tt2/n1240(120)/1513-390 Computationofintercept:bYbt(578,400/15)-3102.5(120/15)=13,74001Equationforlineartrend:Tt=13,740+3102.5tb.1995forecast:Tt=13,740+3102.5(16)=60,277.51996forecast:Tt=13,740+3102.5(17)=66,482.536.a.Agraphofthesedatashowsalineartrend.bThefollowingvaluesareneededtocomputetheslopeandintercept:2ttYt1555200Y750ttComputationofslope:tY()tY/n750(15)(200)/5ttb15122tt2/n55(15)/5Computationofintercept:bYbt40-15(3)=-501Equationforlineartrend:Tt=-5+15tConclusion:averageincreaseinsalesis15unitsperyear37.a.Yes,alineartrendappearstoexist.bThefollowingvaluesareneededtocomputetheslopeandintercept:2ttYt28140595Y2815ttComputationofslope:tY()tY/n2815(28)(595)/7ttb15.5357122tt2/n140(28)/7Computationofintercept:bYbt85-15.5357(4)=22.85701Equationforlineartrend:Tt=22.857+15.536tc.Forecast:T8=22.857+15.536(8)=147.1538.a.Alineartrendappearstobeappropriate.b.T2=12,899.98+2092.066t13-391 c.$2092.066or$2,092,066d.1997:T13=12,899.98+2092.066(13)=40,096.838or$40,096,8381998:T14=12,899.98+2092.066(14)=42,188.904or$42,188,90439.Alineartrenddoesnotseemappropriate.Theplotindicatessometypeofcurvilinearrelationshipovertimesuchas2T=b+bt+btt01t2t16000XNumberX8000XXXXXX01.63.24.86.48.0Year40.a.CenteredSeasonal-IrregulartSalesMovingAverageComponent162153109.2501.0814410.1250.39551011.1250.89961812.1251.48571513.0001.1548714.5000.48391416.5000.848102618.1251.434112319.3751.187121220.2500.593131920.7500.916142821.7501.287152522.8751.093161824.0000.750172225.1250.876183425.8751.314192826.5001.057202127.0000.778212427.5000.873223627.6251.303233028.0001.07113-392 242029.0000.690252830.1250.929264031.6251.26527352827b.Seasonal-IrregularSeasonalQuarterComponentValuesIndex10.899,0.848,0.916,0.876,0.873,0.9290.89021.485,1.434,1.287,1.314,1.303,1.2651.34831.081,1.154,1.187,1.093,1.057,1.0711.10740.395,0.483,0.593,0.750,0.778,0.6900.615Total3.960QuarterAdjustedSeasonalIndex10.89921.36231.11840.621Note:Adjustmentforseasonalindex=4.00/3.96=1.0101c.HudsonMarineexperiencesthelargestseasonalincreaseinquarter2.Sincethisquarteroccurspriortothepeaksummerboatingseason,thisresultseemsreasonable.41.a.CenteredSeasonal-IrregulartSalesMovingAverageComponent1422313.2500.308453.7501.333564.3751.371645.8750.681747.5000.5338147.8751.7789107.8751.2701038.2500.3641158.7500.57112169.7501.641131210.7501.11614911.7500.76615713.2500.528162214.1251.558171815.0001.200181017.3750.5761913203513-393 Seasonal-IrregularSeasonalQuarterComponentValuesIndex11.371,1.270,1.116,1.2001.23920.681,0.364,0.776,0.5760.59730.308,0.533,0.571,0.5280.48541.333,1.778,1.641,1.5581.578Total3.899QuarterAdjustedSeasonalIndex11.27120.61330.49841.619Note:Adjustmentforseasonalindex=4/3.899=1.026b.Thelargesteffectisinquarter4;thisseemsreasonablesinceretailsalesaregenerallyhigherduringOctober,November,andDecember.42.a.Note:Tosimplifythecalculationstheseasonalindexescalculatedinproblem40havebeenroundedtotwodecimalplaces.SeasonalFactorDeseasonalizedSalesYearQuarterSalesYtStYt/St=TtIt1160.906.672151.3611.033101.128.93440.626.4521100.9011.112181.3613.243151.1213.39470.6211.2931140.9015.562261.3619.123231.1220.544120.6219.3541190.9021.112281.3620.593251.1222.324180.6229.0351220.9024.442341.3625.003281.1225.004210.6233.8761240.9026.672361.3626.473301.1226.794200.6232.2671280.9031.112401.3629.413351.1231.254270.6243.5513-394 tYt2(deseasonalized)tYtt16.676.671211.0322.06438.9326.79946.4525.8016511.1155.5525613.2479.4436713.3993.7349811.2990.3264915.56140.04811019.12191.201001120.54225.941211219.35232.201441321.11274.431691420.59288.261961522.32334.802251629.03464.482561724.44415.482891825.00450.003241925.00475.003612033.87677.404002126.67560.074412226.47582.344842326.79616.175292432.26774.245762531.11777.756252629.41764.666762731.25843.757292843.551,219.40784406605.5510,707.347,714tYbbT14.5=21.6271.0556.3296.3291.055t10tb/c.tTrendForecast2936.923037.983139.033240.09YearQuarterTrendForecastSeasonalIndexQuarterlyForecast8136.920.9033.23237.981.3651.65329.031.1243.7113-395 440.090.6224.8643.aNote:Tosimplifythecalculationstheseasonalindexesinproblem40havebeenroundedtotwodecimalplaces.SeasonalFactorDeseasonalizedSalesYearQuarterSalesYtStYt/St=TtIt1141.273.15220.613.28310.502.00451.623.092161.274.72240.616.56340.508.004141.628.6431101.277.87230.614.92350.5010.004161.629.8841121.279.45290.6114.75370.5014.004221.6213.5851181.2714.172100.6116.393130.5026.004351.6221.60Yt2t(deseasonalized)tYtt13.153.15123.286.56432.006.00943.0912.361654.7223.602566.5639.363678.0056.004988.6469.126497.8770.8381104.9249.201001110.00110.00121129.88118.56144139.45122.851691414.75206.501961514.00210.002251613.58217.282561714.17240.8928913-396 1816.39295.023241926.00494.003612021.60432.00400210202.052783.282870tY10.510.1025bbT.995.345.345.995t10tb.yTrendForecast2120.552221.552322.542423.54c.TrendSeasonalQuarterlyYearQuarterForecastIndexForecast6120.551.2726.10221.550.6113.15322.540.5011.27423.541.6238.13Chapter19NonparametricMethodsLearningObjectives1.Learnthedifferencebetweenparametricandnonparametricmethods.2.Knowtheparticularadvantagesofnonparametricmethodsandwhentheyareandwhentheyarenotapplicable.3.Learnhowtousethesigntestfortheanalysisofpairedcomparisons.4.Beabletousethesigntesttoconducthypothesistestsaboutamedian.5.BeabletousetheWilcoxonsigned-ranktestandtheMann-Whitney-Wilcoxontesttodeterminewhetherornottwopopulationshavethesamedistribution.6.BeabletousetheKruskal-Wallistestsforthecomparisonofkpopulations.7.BeabletocomputetheSpearmanrankcorrelationcoefficientandtestforasignificantcorrelationbetweentwosetsofrankings.13-397 13-398 Solutions:1.BinomialProbabilitiesforn=10,p=.50.xProbabilityxProbability0.00106.20511.00987.11722.04398.04393.11729.00984.205110.00105.2461P(0)+P(1)=.0108;addingP(2),exceeds.025requiredinthetail.Therefore,rejectH0ifthenumberofplussignsislessthan2orgreaterthan8.Numberofplussignsis7.DonotrejectH0;concludethatthereisnoindicationthatadifferenceexists.2.Therearen=27casesinwhichavaluedifferentfrom150isobtained.Usethenormalapproximationwithµ=np=.5(27)=13.5and.25n.25(27)2.6Usex=22asthenumberofplussignsandobtainthefollowingteststatistic:x2213.5z3.272.6Withwerejectifz>2.33;sincez=3.27>2.33werejectH0.Conclusion:themedianisgreaterthan150.3.a.Letp=probabilitythesharesheldwillbeworthmoreafterthesplitH0:p.50Ha:p>.50IfH0cannotberejected,thereisnoevidencetoconcludestocksplitscontinuetoaddvaluetostockholdings.b.Letxbethenumberofplussigns(increasesinvalue).Usethebinomialprobabilitytableswithn=18(therewere2tiesinthe20cases)P(x>12)=.0482RejectH0ifthenumberof+signsisgreaterthan12.c.Withx=14,werejectH0.Theresultssupporttheconclusionthatstocksplitsarebeneficialforshareholders.13-399 4.Weneedtodeterminethenumberwhosaidbetterandthenumberwhosaidworse.Thesumofthetwoisthesamplesizeusedforthestudy.n=.34(1253)+.29(1253)=789.4Usethelargesampletestusingthenormaldistribution.Thismeansthevalueofn(n=789.4above)neednotbeinteger.Hence,µ=.5n=.5(789.4)=394.7.25n.25(789.4)14.05Letp=proportionofadultswhofeelchildrenwillhaveabetterfuture.H0:p.50Ha:p>.50Withx=.34(1253)=426x426394.7z2.2314.05With=.05,werejectH0ifz>1.645Sincez=2.23>1.645,werejectH0Conclusion:morethanhalfoftheadultsfeeltheirchildrenwillhaveabetterfuture.5.n=185+165=350µ=0.5n=0.5(350)=175.25n.25(350)9.35RejectH0ifz<-1.96orifz>1.96185175z1.079.35DonotrejectH0;cannotconcludethereisadifferenceinpreferenceforthetwoshows.6.n=202+158=360µ=0.5n=0.5(360)=180.25n.25(360)9.49RejectH0ifz<-1.96orifz>1.96202180z2.329.4913-400 RejectH0;concludePackardBellandCompaqhavedifferentmarketshares.7.µ=0.5n=0.5(300)=150.25n.25(300)8.66165150z1.738.66p-value=2(.5000-.4582)=.0836DonotrejectH0;weareunabletoconcludethatthemedianannualincomediffers.8.µ=.5n=.5(150)=75.25n.25(150)6.12Onetailedtest:rejectH0ifz>1.645For98+signs9875z3.766.12RejectH0;concludethatahometeamadvantageexists.9.H0:Median15Ha:Median>15Usebinomialprobabilitieswithn=8andp=.50:Onetailtestwith=.05,P(8+’s)=.0039P(7+’s)=.0312.0351P(6+’s)=.1094RejectH0if7or8+’s.With7+’sinthesample,rejectH0.Datadoesenableustoconcludethattherehasbeenanincreaseinthemediannumberofpart-timeemployees.10.H0:Median=152Ha:Median152µ=.5n=.5(225)=112.5.25n.25(225)7.513-401 RejectH0ifz<-1.96orifz>1.96For122cases122112.5z1.277.5DonotrejectH0;weareunabletoconcludethatthemedianannualincomeneededdiffersfromthatreportedinthesurvey.11.n=50µ=0.5n=0.5(50)=25.25n.25(50)3.54RejectH0ifz>1.64533hadwagesgreaterthan$5853325z2.263.54RejectH0;concludethatthemedianweeklywageisgreaterthan$585.12.H0:ThepopulationsareidenticalHa:ThepopulationsarenotidenticalAdditive1Additive2DifferenceAbsoluteValueRankSignedRank20.1218.052.072.079+923.5621.771.791.797+722.0322.57-.54.543-319.1517.062.092.0910+1021.2321.22.01.011+124.7723.80.97.974+416.1617.20-1.041.045-518.5514.983.573.5712+1221.8720.031.841.848+824.2321.153.083.0811+1123.2122.78.43.432+225.0223.701.321.326+6Total62µT=0nn(1)(2n1)12(13)(25)25.5T66T620Tz2.4325.5T13-402 Two-tailedtest.RejectH0ifz<-1.96orifz>1.96Sincez=2.43>1.96werejectH0.Conclusion:thereisasignificantdifferenceintheadditives.13.WithoutWithRankofAbsoluteRelaxantRelaxantDifferenceDifferenceSignedRank151059912102332212101010811-36.5-6.510911175233810-23-310736.56.5141136.56.59636.56.5Total36µT=0nn(1)(2n1)10(11)(21)19.62T66T36Tz1.8319.62TOne-tailedtest.RejectH0ifz>1.645Sincez=1.83>1.645werejectH0.Conclusion:thereisasignificantdifferenceinfavoroftherelaxant.14.AbsoluteAirportDifferenceDifferenceSignedRankBostonLogan0.190.1910ChicagoMidway-0.020.02-3.5ChicagoO"Hare0.050.056Denver0.040.045FortLauderdale-0.010.01-1.5LosAngeles0.060.067Miami0.020.023.5NewYork(JFK)0.090.098OrangeCounty(CA)0.160.169Washington(Dulles)0.010.011.5T=45nn(1)(2n1)10(11)(21)19.62T6613-403 T450Tz2.2919.62TRejectH0ifz<-1.96orifz>1.96.RejectH0;concludeadifferenceexistswithAvishigher.15.RankofAbsoluteService#1Service#2DifferenceDifferenceSignedRank24.528.0-3.57.5-7.526.025.50.51.51.528.032.0-4.09.5-9.521.020.01.044.018.019.5-1.56-6.036.028.08.01111.025.029.0-4.09.5-9.521.022.0-1.04-4.024.023.50.51.51.526.029.5-3.57.5-7.531.030.01.044.0T=-22.0µT=0nn(1)(2n1)11(12)(23)22.49T66T22Tz.9822.49TRejectH0ifz<-1.96orifz>1.96.Sincez=-.98,donotrejectH0;thereisnosignificantdifference.16.1997P/ERatioEst.1998P/ERatioDifferenceRankSignedRank4032899242222.52.52123-22.5-2.53023788251966.56.519190002017344291910101035201511111718-11-1332766.56.52016455T=59n=11(discardingthe0)µT=013-404 nn(1)(2n1)11(12)(23)22.49T66RejectH0ifz<-1.96orifz>1.96590z2.6222.49RejectH0;concludeadifferenceexistsbetweenthe1997P/Eratiosandtheestimated1998P/Eratios.17.RankofAbsolutePrecampaignPostcampaignDifferenceDifferenceSignedRank130160-3010-10100105-52.5-2.5120140-209-9959052.52.5140130104.54.58082-21-16555104.54.590105-157.5-7.5140152-126-6125140-157.5-7.5T=-32µT=0nn(1)(2n1)10(11)(21)19.62T66T32Tz1.6319.62TRejectH0ifz<-1.63DonotrejectH0;thedifferenceisnotsignificantatthe=.05level.18.Rankthecombinedsamplesandfindtheranksumforeachsample.Thisisasmallsampletestsincen1=7andn2=9Additive1Additive2MPGRankMPGRank17.3218.78.518.4617.8419.11021.31516.7121.01418.2522.11618.6718.78.517.5319.8113420.71320.21213-405 102T=34With=.05,n1=7andn2=9TL=41andTU=7(7+9+1)-41=78SinceT=34<41,werejectH0Conclusion:thereisasignificantdifferenceingasolinemileage19.a.PublicFinancialAccountantRankPlannerRank25.2524.0233.81924.2331.31628.11033.21830.91529.21326.98.530.01428.61125.9624.7434.52028.91231.71726.8726.98.523.91136.573.511nnn(1)10(10101)105T11222T=136.511nnn(n1)(10)(10)(21)13.23T12121212RejectH0ifz<-1.96orifz>1.96136.5105z2.3813.23RejectH0;salariesdiffersignificantlyforthetwoprofessions.b.PublicAccountant:$30,200FinancialPlanner:$26,700Conclusion:thereisasignificantdifferenceinstartingsalaries20.a.Median4thsalaryforeachMen49.9Women35.4b.MenRankWomenRank30.6444.5813-406 75.51435.4545.2927.9362.21340.5738.2625.8249.91147.51055.31224.81T=36FromTablesTL=37T1.645Sincez=2.05>1.645werejectH0Conclusion:thereisasignificantdifferencebetweenthepopulations.22.H0:ThereisnodifferenceinthedistributionsofP/EratiosHa:ThereisadifferencebetweenthedistributionsofP/EratiosWewillrejectH0ifz<-2.33orz>2.33JapanUnitedStatesCompanyP/ERatioRankCompanyP/ERatioRankSumitomoCorp.15320Gannet196Kinden218Motorola2411.5Heiwa185Schlumberger2411.5NCRJapan12519OracleSystems4316SuzukiMotor3113Gap2210FujiBank21321Winn-dixie142SumitomoChemical6417Ingersoll-Rand218SeibuRailway66622Am.Elec.Power142Shiseido3314Hercules218TohoGas6818TimesMirror3815Total157WellPointHealth154No.StatesPower14213-407 Total9611nnn(1)10(10121)115T1122211nnn(n1)(10)(12)(10121)15.17T12121212T=157157115z2.7715.17Sincez=2.77>2.33,rejectH0.WeconcludethatthereisasignificantdifferenceinP/Eratiosforthetwocountries.23.Sumofranks(Winter)=71.5Sumofranks(Summer)=138.5UseT=71.511nnn(1)10(21)105T1122211nnn(n1)(10)(10)(21)13.23T12121212T71.5105Tz2.5313.23TRejectH0ifz<-1.96orifz>1.96RejectH0;thereisasignificantdifference24.Sumofranks(Dallas)=116Sumofranks(SanAntonio)=160UseT=11611nnn(1)10(24)120T1122211nnn(n1)(10)(13)(24)16.12T12121212T116120Tz.2516.12TRejectH0ifz<-1.96orifz>1.96DonotrejectH0;thereisnotsignificantevidencetoconcludethatthereisadifference.13-408 25.KitchenRankMasterRankBedroom25,2001618,000417,400222,9001122,8001026,4001721,900924,8001519,7005.526,9001823,0001217,800319,7005.524,6001416,900121,000721,80088923,6001382FromAppendixB,TL=73TU=n1(n1+n2+1)-TL=10(10+8+1)-73=117RejectH0ifT<73orifT>117SinceT=82,donotrejectThereisnosignificantdifferencebetweenthecosts.26.ABC41178142101513126913534652122212(34)(65)(21)W3(16)10.22(15)(16)5552=5.99147(2degreesoffreedom).05RejectH0;concludethattheratingsfortheproductsdiffer.27.13-409 ABC11.55.017.02.511.520.08.02.515.010.04.08.08.06.016.018.01.019.013.014.058.043.0109.022212(58)(43)(109)W3(21)9.06(20)(21)6772=9.21034(2degreesoffreedom).01DonotrejectH0;wecannotconcludethatthereisasignificantdifferenceintestpreparationprograms.28.SwimmingRankTennisRankCyclingRank408841593855380448514250142511450132953400642010402742712530152682Sum41611822212(41)(61)(18)W3(151)9.26(15)(151)5552=5.99147.05Since9.26>5.99147,rejectH0;concludethatthereisasignificantdifferenceincaloriesamongthethreeactivities.29.ABC221274.5144.5910.52713710.51522.53364.522212(22.5)(33)(64.5)W3(16)9.555(15)(16)5552=5.99147(2degreesoffreedom).0513-410 Since9.555>5.99147werejectH0andconcludethatthereisasignificantdifferenceingasmileageamongthethreeautomobiles.30.Course1Course2Course3Course4321920147164101915125186131117852267953222212(52)(26)(79)(53)W3(21)8.03(20)(21)55552=7.81473(3degreesoffreedom).05Since8.03>7.81473,werejectH0andconcludethatthereisasignificantdifferenceinthequalityofcoursesofferedbythefourmanagementdevelopmentcenters.31.M&MsKitKatMilkyWayII10.593756131441512210.58156481622212(56)(48)(16)W3(16)8.96(15)(16)5552=5.99147(2degreesoffreedom).05Since8.96>5.99147werejectH0Therearesignificantdifferencesincaloriecontentamongthethreecandies.232.a.d=52i26d6(52)ir11.68s2nn(1)10(99)11b..33rsn19r0.68sz2.06.33rsRejectifz<-1.96orifz>1.9613-411 Sincez=2.06>1.96,werejectH0.Concludethatsignificantrankcorrelationexists.33.Case1:2d=0i26d6(0)ir111s2nn(1)6(361)Case2:2d=70i26d6(70)ir111s2nn(1)6(361)Withperfectagreement,rs=1.Withexactoppositeranking,rs=-1.234.d=250i26d6(250)ir11.136s2nn(1)11(120)11.32rsn110r0.136sz.425.32rsRejectifz<-1.96orifz>1.96Sincez=-.425,wecannotrejectH0.Concludethatthereisnotasignificantrelationshipbetweentherankings.235.a.d=54i26d6(54)ir11.6722nn(1)10(101)b.H0:pr0Ha:pr>013-412 11.3333rsn1101rsr.670zs2.02.3333rsp-value=.5000-.4783=.0217c.RejectH0:Concludeasignificantpositiverankcorrelation.36.DrivingDistancePuttingdid2i15-41656-11410-6369274967-1110374928-63639-6367439817492d=282i26d6(282)ir11.709s2nn(1)10(1001)=0rs11.333rsn19RejectH0ifz<-1.645orifz>1.645.7090z2.13.333RejectH0;thereisasignificantnegativerankcorrelationbetweendrivingdistanceandputting.237.d=38i26d6(38)ir11.77s2nn(1)10(99)=0rs13-413 11.3333rsn19RejectH0ifz<-1.645orifz>1.645r0.77sz2.31.3333rsRejectH0;thereisasignificantrankcorrelationbetweencurrentstudentsandrecentgraduates.38.n=905+1045=1950=.5n=.5(1950)=975.25n.25(1950)22.01RejectH0ifz<-1.96orifz>1.96905975z3.1722.01RejectH0;thedifferenceinthefavor-opposeopinionissignificant.39.a.n=11+32=43H0:Median$118,000Ha:Median<$118,000µ=.5n=.5(43)=21.5.25n.25(43)3.2787x1121.5z3.203.2787Sincez=-3.20<-1.645,rejectH0.WeconcludethatthemedianresalepriceforhomesinHouston,Texasisbelowthenationalmedian.b.n=27+13=40H0:Median$118,000Ha:Median>$118,000µ=.5n=.5(40)=20.25n.25(40)3.1623x2720z2.213.162313-414 Sincez=2.21>1.645,rejectH0.WeconcludethatthemedianresalepriceforhomesinBoston,Massachusettsisabovethenationalmedian.40.UsetheWilcoxonSignedRankTestHomemakerDifferenceSignedRank1-250-11240235034-150-65-330-126-180-77-190-8.58-230-109-100-510-190-8.511-90-412201T=-66=0Tnn(1)(2n1)12(13)(25)25.5T66RejectH0ifz<-1.96orifz>1.96T66Tz2.5925.5TRejectH0;concludethatthemodelsdifferintermsofsellingprices.41.RankofDifferenceAbsoluteDifferenceSignedRank1.51010.01.299.0-.22.5-2.513-415 0.544.0.766.0.877.01.088.00.655.0.22.52.5-.011-1.0T=48nn(1)(2n1)10(11)(21)19.62T66RejectH0ifz>1.645T48Tz2.4519.62TRejectH0;concludethatthereisasignificantweightgain.42.UsetheMWWtest.Sumofranks(line1)=70Sumofranks(line2)=183T=7011nnn(1)10(23)115T1122211nnn(n1)(10)(12)(23)15.17T12121212RejectH0ifz<-1.645orifz>1.645T70115Tz2.9715.17TRejectH0;concludethattheweightsdifferforthetwoproductionlines.43.Method1Method2Method38.54.52.015.014.07.06.016.010.017.08.51.018.012.53.012.511.04.513-416 77.066.527.522212(77)(66.5)(27.5)W3(19)7.956(18)(19)6662=5.99147(3degreesoffreedom).05Since7.956>5.99147,werejectH0andconcludethatthereisasignificantdifferenceamongthemethods.44.NoProgramCompanyProgramOffSiteProgram1612792011017415192116313188145741063022212(74)(106)(30)W3(21)12.61(20)(21)6772=7.37776(2degreesoffreedom).05Since12.61>7.37776,werejectH0;thereisasignificantdifferenceamongtheprograms.45.BlackJenningsSwansonWilson22.520.522.59.59.527.06.017.58.07.02.51.02.517.512.55.026.028.517.524.04.028.512.520.517.515.025.014.011.072.5171.5113.577.513-417 222212(72.5)(171.5)(113.5)(77.5)W3(30)6.344(29)(30)68962=7.81473(3degreesoffreedom).05Since6.344<7.81473wecannotrejectH0.Wecannotconcludethatthereisasignificantdifferenceamongthecourseevaluationratingsforthe4instructors.246.d=136i26d6(136)ir11.76s2nn(1)15(224)11.2673rsn114RejectH0ifz<-1.645orifz>1.645rsr.76zs2.84.2673rsRejectH0;concludethatthereisasignificantrankcorrelationbetweenthetwoexams.Chapter20StatisticalMethodsforQualityControlLearningObjectives1.Learnabouttheimportanceofqualitycontrolandhowstatisticalmethodscanassistinthequalitycontrolprocess.2.Learnaboutacceptancesamplingprocedures.3.Knowthedifferencebetweenconsumer’sriskandproducer’srisk.4.Beabletousethebinomialprobabilitydistributiontodevelopacceptancesamplingplans.5.Knowwhatismeantbymultiplesamplingplans.6.Beabletoconstructqualitycontrolchartsandunderstandhowtheyareusedforstatisticalprocesscontrol.7.Knowthedefinitionsofthefollowingterms:13-418 producer"sriskassignablecausesconsumer"sriskcommoncausesacceptancesamplingcontrolchartsacceptablecriterionuppercontrollimitoperatingcharacteristiccurvelowercontrollimit13-419 Solutions:1.a.Forn=4UCL=+3(/n)=12.5+3(.8/4)=13.7LCL=-3(/n)=12.5-3(.8/4)=11.3b.Forn=8UCL=+3(.8/8)=13.35LCL=-3(.8/8)=11.65Forn=16UCL=+3(.8/16)=13.10LCL=-3(.8/16)=11.90c.UCLandLCLbecomeclosertogetherasnincreases.Iftheprocessisincontrol,thelargersamplesshouldhavelessvarianceandshouldfallcloserto12.5.6775.2.a.542.255()b.UCL=+3(/n)=5.42+3(.5/5)=6.09LCL=-3(/n)=5.42-3(.5/5)=4.751353.a.p00540.25100()pp().(10054009460.)b.00226.pn100c.UCL=p+3=0.0540+3(0.0226)=0.1218pLCL=p-3=0.0540-3(0.0226)=-0.0138pUseLCL=04.RChart:UCL=RD=1.6(1.864)=2.984LCL=RD=1.6(0.136)=0.223xChart:UCL=xAR=28.5+0.373(1.6)=29.102LCL=xAR=28.5-0.373(1.6)=27.9025.a.UCL=+3(/n)=128.5+3(.4/6)=128.99LCL=-3(/n)=128.5-3(.4/6)=128.011-420 DataandStatistics7724.b.xxn/12873.incontroli67743.c.xxn/12905.outofcontroli620121990..6.ProcessMean=2001.2UCL=+3(/n)=20.01+3(/5)=20.12Solvefor:(.201220015.)0082.37.SampleNumberObservationsxiRi131422833.6714226183526.3317325303429.679417252121.008538293534.009641423639.676721172922.3312832262828.676941343336.0081029173025.33131126314032.33141223192522.3361317243224.33151443351731.67261518252924.00111630423134.33121728363232.0081840293133.33111918292825.00112022342627.3312R=11.4andx2917.RChart:UCL=RD=11.4(2.575)=29.354LCL=RD=11.4(0)=03xChart:UCL=xAR=29.17+1.023(11.4)=40.82LCL=xAR=29.17-1.023(11.4)=17.521-421 Chapter1RChart:30UCL=29.320R=11.4100LCL=01234567891011121314151718192016SampleNumberxChart:UCL=40.840=30x=29.1720LCL=17.51234567891011121314151617181920SampleNumber1418.a.p00470.20150()pp().(10047009530.)b.00173.pn150UCL=p+3=0.0470+3(0.0173)=0.0989pLCL=p-3=0.0470-3(0.0173)=-0.0049p1-422 DataandStatisticsUseLCL=012c.p008.150Processshouldbeconsideredincontrol.d.p=.047,n=150UCL=np+3np()1p=150(0.047)+315000470953(.)(.)=14.826LCL=np-3np()1p=150(0.047)-315000470953(.)(.)=-0.726Thus,theprocessisoutofcontrolifmorethan14defectivepackagesarefoundinasampleof150.e.Processshouldbeconsideredtobeincontrolsince12defectivepackageswerefound.f.Thenpchartmaybepreferredbecauseadecisioncanbemadebysimplycountingthenumberofdefectivepackages.9.a.Totaldefectives:165165p00413.20200()pp().(10041309587.)b.00141.pn200UCL=p+3=0.0413+3(0.0141)=0.0836pLCL=p-3=0.0413+3(0.0141)=-0.0010pUseLCL=020c.p010.Outofcontrol200d.p=.0413,n=200UCL=np+3np()1p=200(0.0413)+32000041309587(.)(.)=16.702LCL=np-3np()1p=200(0.0413)-32000041309587(.)(.)=0.1821e.Theprocessisoutofcontrolsince20defectivepistonswerefound.n!xnx10.fx()pp()1xnx!()!Whenp=.02,theprobabilityofacceptingthelotis1-423 Chapter125!025f()0(.)(0021002.)06035.0!(250)!Whenp=.06,theprobabilityofacceptingthelotis25!025f()0(.)(0061006.)02129.0!(250)!11.a.Usingbinomialprobabilitieswithn=20andp0=.02.P(Acceptlot)=f(0)=.6676Producer’srisk:=1-.6676=.3324b.P(Acceptlot)=f(0)=.2901Producer’srisk:=1-.2901=.709912.Atp0=.02,then=20andc=1planprovidesP(Acceptlot)=f(0)+f(1)=.6676+.2725=.9401Producer’srisk:=1-.9401=.0599Atp0=.06,then=20andc=1planprovidesP(Acceptlot)=f(0)+f(1)=.2901+.3703=.6604Producer’srisk:=1-.6604=.3396Foragivensamplesize,theproducer’sriskdecreasesastheacceptancenumbercisincreased.13.a.Usingbinomialprobabilitieswithn=20andp0=.03.P(Acceptlot)=f(0)+f(1)=.5438+.3364=.8802Producer’srisk:=1-.8802=.1198b.Withn=20andp1=.15.P(Acceptlot)=f(0)+f(1)=.0388+.1368=.1756Consumer’srisk:=.1756c.Theconsumer’sriskisacceptable;however,theproducer’sriskassociatedwiththen=20,c=1planisalittlelargerthandesired.1-424 DataandStatistics14.P(Accept)Producer’sP(accept)Consumer’scp0=.05Riskp1=.30Risk(n=10)0.5987.4013.0282.02821.9138.0862.1493.14932.9884.0116.3828.3828(n=15)0.4633.5367.0047.00471.8291.1709.0352.03522.9639.0361.1268.12683.9946.0054.2968.2968(n=20)0.3585.6415.0008.00081.7359.2641.0076.00762.9246.0754.0354.03543.9842.0158.1070.1070Theplanwithn=15,c=2isclosewith=.0361and=.1268.However,theplanwithn=20,c=3isnecessarytomeetbothrequirements.15.a.P(Accept)shownforpvaluesbelow:cp=.01p=.05p=.08p=.10p=.150.8179.3585.1887.1216.03881.9831.7359.5169.3918.17562.9990.9246.7880.6770.4049TheoperatingcharacteristiccurveswouldshowtheP(Accept)versuspforeachvalueofc.b.P(Accept)cAtp0=.01Producer’sRiskAtp1=.08Consumer’sRisk0.8179.1821.1887.18871.9831.0169.5169.51692.9990.0010.7880.7880x190816.a.954.2020b.UCL=+3(/n)=95.4+3(.50/5)=96.07LCL=-3(/n)=95.4-3(.50/5)=94.73c.No;allwereincontrol1-425 Chapter117.a.Forn=10UCL=+3(/n)=350+3(15/10)=364.23LCL=-3(/n)=350-3(15/10)=335.77Forn=20UCL=350+3(15/20)=360.06LCL=350-3(15/20)=339.94Forn=30UCL=350+3(15/30)=358.22LCL=350-3(15/30)=343.78b.Bothcontrollimitscomeclosertotheprocessmeanasthesamplesizeisincreased.c.Theprocesswillbedeclaredoutofcontrolandadjustedwhentheprocessisincontrol.d.Theprocesswillbejudgedincontrolandallowedtocontinuewhentheprocessisoutofcontrol.e.Allhavez=3wherearea=.4986P(TypeI)=1-2(.4986)=.002818.RChart:UCL=RD=2(2.115)=4.234LCL=RD=2(0)=03xChart:UCL=xAR=5.42+0.577(2)=6.572LCL=xAR=5.42-0.577(2)=4.272EstimateofStandardDeviation:R2086.d2326.219.R=0.665x=95.398xChart:UCL=xAR=95.398+0.577(0.665)=95.7822LCL=xAR=95.398-0.577(0.665)=95.0142RChart:UCL=RD=0.665(2.115)=1.40641-426 DataandStatisticsLCL=RD=0.665(0)=03TheRchartindicatedtheprocessvariabilityisincontrol.Allsamplerangesarewithinthecontrollimits.However,theprocessmeanisoutofcontrol.Sample11(x=95.80)andSample17(x=94.82)falloutsidethecontrollimits.20.R=.053x=3.082xChart:UCL=xAR=3.082+0.577(0.053)=3.1122LCL=xAR=3.082-0.577(0.053)=3.0512RChart:UCL=RD=0.053(2.115)=0.11214LCL=RD=0.053(0)=03Alldatapointsarewithinthecontrollimitsforbothcharts.21.a..08UCL.06.04.02LCL0Warning:Processshouldbechecked.Allpointsarewithincontrollimits;however,allpointsarealsogreaterthantheprocessproportiondefective.1-427 Chapter1b.25UCL2423LCL22Warning:Processshouldbechecked.AllpointsarewithincontrollimitsyetthetrendinpointsshowamovementorshifttowardUCLout-of-controlpoint.22.a.p=.04pp().(10040.)9600139.pn200UCL=p+3=0.04+3(0.0139)=0.0817pLCL=p-3=0.04-3(0.0139)=-0.0017pUseLCL=0b.1-428 DataandStatisticsoutofcontrolUCL(.082).04LCL(0)Formonth1p=10/200=0.05.Othermonthlyvaluesare.075,.03,.065,.04,and.085.Onlythelastmonthwithp=0.085isanout-of-controlsituation.23.a.Usebinomialprobabilitieswithn=10.Atp0=.05,P(Acceptlot)=f(0)+f(1)+f(2)=.5987+.3151+.0746=.9884Producer’sRisk:=1-.9884=.0116Atp1=.20,P(Acceptlot)=f(0)+f(1)+f(2)=.1074+.2684+.3020=.6778Consumer’srisk:=.6778b.Theconsumer’sriskisunacceptablyhigh.Toomanybadlotswouldbeaccepted.c.Reducingcwouldhelp,butincreasingthesamplesizeappearstobethebestsolution.24.a.P(Accept)areshownbelow:(Usingn=15)p=.01p=.02p=.03p=.04p=.05f(0).8601.7386.6333.5421.4633f(1).1303.2261.2938.3388.3658.9904.9647.9271.8809.8291=1-P(Accept).0096.0353.0729.1191.1709Usingp0=.03sinceiscloseto.075.Thus,.03isthefractiondefectivewheretheproducerwilltoleratea.075probabilityofrejectingagoodlot(only.03defective).b.p=.25f(0).0134f(1).06681-429 Chapter1=.080225.a.P(Accept)whenn=25andc=0.Usethebinomialprobabilityfunctionwithn!xnxfx()pp()1xnx!()!or25!02525fp()0()()11pp0!25!Iff(0)p=.01.7778p=.03.4670p=.10.0718p=.20.0038b.1.0.8.6P(Accept).4.2.00.02.04.06.08.10.12.14.16.18.20PercentDefectivec.1-f(0)=1-.778=.22226.a.=np=250(.02)=5np()1p250002098(.)(.).221P(Accept)=P(x10.5)1055.z249.221.P(Accept)=.5000+.4936=.9936Producer’sRisk:=1-.9936=.00641-430 DataandStatisticsb.=np=250(.08)=20np()1p250008092(.)(.).429P(Accept)=P(x10.5)1055.z221.429.P(Accept)=1-.4864=.0136Consumer’sRisk:=.0136c.Theadvantageistheexcellentcontrolovertheproducer’sandtheconsumer’srisk.Thedisadvantageisthecostoftakingalargesample.Chapter21SampleSurveyLearningObjectives1.Learnwhatasamplesurveyisandhowitdiffersfromanexperimentasamethodofcollectingdata.2.Knowaboutthemethodsofdatacollectionforasurvey.3.Knowthedifferencebetweensamplingandnonsamplingerror.4.Learnaboutfoursampledesigns:(1)simplerandomsampling,(2)stratifiedsimplerandomsampling,(3)clustersampling,and(4)systematicsampling.5.Leanhowtoestimateapopulationmean,apopulationtotal,andapopulationproportionusingtheabovesampledesigns.6.Understandtherelationshipbetweensamplesizeandprecision.7.Learnhowtochoosetheappropriatesamplesizeusingstratifiedandsimplerandomsampling.8.Learnhowtoallocatethetotalsampletothevariousstratausingstratifiedsimplerandomsampling.1-431 Chapter1Solutions:1.a.x=215isanestimateofthepopulationmean.2080050b.s2.7386x50800c.2152(2.7386)or209.5228to220.47722.a.Estimateofpopulationtotal=Nx=400(75)=30,000b.EstimateofStandardError=Nsx840080Ns400320x80400c.30,0002(320)or29,360to30,6401-432 DataandStatistics3.a.p=.30isanestimateofthepopulationproportion1000100(.3)(.7)b.s.0437p100099c..302(.0437)or.2126to.38744.B=152(70)4900n72.983022(15)(70)67.13894450Asamplesizeof73willprovideanapproximate95%confidenceintervalofwidth30.5.a.x=149,670ands=73,4207715073,420s10,040.83x77150approximate95%confidenceinterval149,6702(10,040.83)or$129,588.34to$169,751.66b.X=Nx=771(149,670)=115,395,570sx=Nsx=771(10,040.83)=7,741,479.93approximate95%confidenceinterval115,395,7702(7,741,479.93)or$99,912,810.14to$130,878,729.8677150(.36)(.64)c.p=18/50=0.36ands.0663p77149approximate95%confidenceinterval1-433 Chapter10.362(0.0663)or0.2274to0.4926Thisisaratherlargeinterval;samplesizesmustberatherlargetoobtaintightconfidenceintervalsonapopulationproportion.6.B=5000/2=2500Usethevalueofsforthepreviousyearintheformulatodeterminethenecessarysamplesize.2(31.3)979.69n336.005122(2.5)(31.3)2.91574724Asamplesizeof337willprovideanapproximate95%confidenceintervalofwidthnolargerthan$5000.7.a.Stratum1:=138Stratum2:x=1032Stratum3:x=2103b.Stratum1x=13813020020s6.3640x1202001382(6.3640)or125.272to150.728Stratum2x=10322525030s4.2817x2302501032(4.2817)or94.4366to111.56341-434 DataandStatisticsStratum3x=21035010025s8.6603x3251002102(8.6603)or192.6794to227.3206200250100c.x138103210st550550550=50.1818+46.8182+38.1818=135.18182221(30)(25)(50)s200(180)250(220)100(75)xst2(550)20302513,515,833.33.40922(550)approximate95%confidenceinterval135.18182(3.4092)or128.3634to142.00028.a.Stratum1:Nx=200(138)=27,60011Stratum2:Nx=250(103)=25,75022Stratum3:Nx=100(210)=21,00033b.Nx=27,600+25,750+21,000=74,350stNote:thesumoftheestimateforeachstratumtotalequalsNxstc.Nx=550(3.4092)=1875.06(see7c)stapproximate95%confidenceinterval1-435 Chapter174,3502(1875.06)or70,599.88to78,100.129.a.Stratum1p=.50120020(.50)(.50)s.1088p120019502(.1088)or.2824to.7176Stratum2p=.78225030(.78)(.22)s.0722p225029.782(.0722)or.6356to.9244Stratum3p=.21310025(.21)(.79)s.0720p310024.212(.0720)or.066to.354200250100b.p(.50)(.78)(.21).5745st5505505501(.5)(.5)(.78)(.22)(.21)(.79)c.s200(180)250(220)100(75)pst2(550)1929241-436 DataandStatistics1(473.6842325.448351.8438).05302(550)d.approximate95%confidenceinterval.57452(.0530)or.4685to.68052300(150)600(75)500(100)(140,000)210.a.n92.8359(20)2196,000,00015,125,000222(1400)300(150)600(75)500(100)2Roundingupwechooseatotalsampleof93.300(150)n93301140,000600(75)n93302140,000500(100)n93333140,000b.WithB=10,thefirstterminthedenominatorintheformulafornchanges.22(140,000)(140,000)n305.65302(10)49,000,00015,125,0002(1400)15,125,0004Roundingup,weseethatasamplesizeof306isneededtoprovidethislevelofprecision.300(150)n306981140,000600(75)n306982140,000500(100)n3061093140,000Duetorounding,thetotaloftheallocationstoeachstrataonlyaddto305.Note1-437 Chapter1thateventhoughthesamplesizeislarger,theproportionallocatedtoeachstratumhasnotchanged.22(140,000)(140,000)n274.60602(15,000)56,250,00015,125,00015,125,0004Roundingup,weseethatasamplesizeof275willprovidethedesiredlevelofprecision.Theallocationstothestrataareinthesameproportionasforpartsaandb.300(150)n275981140,000600(75)n275882140,000500(100)n275983140,000Again,duetorounding,thestratumallocationsdonotaddtothetotalsamplesize.Anotheritemcouldbesampledfrom,say,stratum3ifdesired.11.a.x=29.5333x=64.77512x=45.2125x=53.030034b.Indianapolis13.360338629.533263829.53310.9086(.9177)or19.5222to39.5438Louisville25.066645864.775284564.77517.7248(.9068)or48.7022to80.84781-438 DataandStatisticsSt.Louis19.408480845.2125288045.2125(13.7238)(.9487)or32.1927to58.2323Memphis29.6810701053.03002107053.030018.7719(.9258)or35.6510to70.4090381455803705c.p.4269st2336233823382331015pp(1)6611d.NNn()38(32)33.7778111n15153pp(1)8822NNn()45(37)55.7478222n17235pp(1)8833NNn()80(72)192.8571333n17355pp(1)101044NNn()70(60)116.6667444n19411s33.777855.7478192.8571116.6667(399.0494).0857pst22(233)(233)approximate95%confidenceinterval.42692(.0857)or1-439 Chapter1.2555to.598312.a.St.Louistotal=Nx=80(45.2125)=361711Indollars:$3,617,000b.Indianapolistotal=Nx=38(29.5333)=1122.265411Indollars:$1,122,26538458070c.x29.533364.77545.212553.030048.7821st23323323323322s(13.3603)1NNn()38(32)36,175.517111n6122s(25.0666)2NNn()45(37)130,772.1222n8222s(19.4084)3NNn()80(72)271,213.91333n8322s(29.6810)4NNn()70(60)370,003.94444n1041s36,175.517130,772.1271,213.91370,003.94xst2(233)1(808,165.47)3.85832(233)approximate95%confidenceintervalx2sstxst48.78212(3.8583)or41.0655to56.4987Indollars:$41,066to$56,499d.approximate95%confidenceintervalNx2Nsstxst1-440 DataandStatistics233(48.7821)2(233)(3.8583)11,366.2291797.9678or9,568.2612to13,164.197Indollars:$9,568,261to$13,164,197250(80)38(150)35(45)(11,275)213.n27.3394(30)23,404,0251,245,8752222(123)50(80)38(150)35(45)4Roundingupweseethatasamplesizeof28isnecessarytoobtainthedesiredprecision.50(80)n2810111,27538(150)n2814211,27535(45)n284311,2752250(100)38(100)35(100)123(100)b.n3322(30)3,404,025123(100)2222(123)50(100)38(100)35(100)450(100)n3313112,30038(100)n3310212,30035(100)n339312,300Thisisthesameasproportionalallocation.NotethatforeachstratumNhnnhNx750i14.a.x15cM50i1-441 Chapter1XMx=300(15)=4500ca15ip.30cM50ib.2()xxM=[95-15(7)]2+[325-15(18)]2+[190-15(15)]2+[140-15ici(10)]2=(-10)2+(55)2+(-35)2+(-10)2=44502544450s1.4708xc2(25)(4)(12)3sXMsxc=300(1.4708)=441.242()apM=[1-.3(7)]2+[6-.3(18)]2+[6-.3(15)]2+[2-.3(10)]2ici=(-1.1)2+(.6)2+(1.5)2+(-1)2=4.822544.82s.0484pc2(25)(4)(12)3c.approximate95%confidenceIntervalforPopulationMean:152(1.4708)or12.0584to17.9416d.approximate95%confidenceIntervalforPopulationTotal:45002(441.24)or3617.52to5382.48e.approximate95%confidenceIntervalforPopulationProportion:.302(.0484)or.2032to.39681-442 DataandStatistics10,40015.a.80xc130XMx=600(80)=48,000c13p.10c130b.2()xxM=[3500-80(35)]2+[965-80(15)]2+[960-80(12)]2ici+[2070-80(23)]2+[1100-80(20)]2+[1805-80(25)]2=(700)2+(-235)2+(0)2+(230)2+(-500)2+(-195)2=886,150306886,150s7.6861xc2(30)(6)(20)5approximate95%confidenceIntervalforPopulationMean:802(7.6861)or64.6278to95.3722c.s=600(7.6861)=4611.66Xapproximate95%confidenceIntervalforPopulationTotal:48,0002(4611.66)or38,776.68to57,223.322()apM=[3-.1(35)]2+[0-.1(15)]2+[1-.1(12)]2+[4-.1(23)]2ici+[3-.1(20)]2+[2-.1(25)]2=(-.5)2+(-1.5)2+(-.2)2+(1.7)2+(1)2+(-.5)2=6.683066.68s.0211pc2(30)(6)(20)5approximate95%confidence1-443 Chapter1IntervalforPopulationProportion:.102(.0211)or.0578to.1422200016.a.40xc50Estimateofmeanageofmechanicalengineers:40years35b.p.70c50Estimateofproportionattendinglocaluniversity:.702c.()xxM=[520-40(12)]2+···+[462-40(13)]2ici=(40)2+(-7)2+(-10)2+(-11)2+(30)2+(9)2+(22)2+(8)2+(-23)2+(-58)2=7292120107292s2.0683xc2(120)(10)(50/12)9approximate95%confidenceIntervalforMeanage:402(2.0683)or35.8634to44.13662d.()apM=[8-.7(12)]2+···+[12-.7(13)]2ici=(-.4)2+(-.7)2+(-.4)2+(.3)2+(-1.2)2+(-.1)2+(-1.4)2+(.3)2+(.7)2+(2.9)2=13.31201013.3s.0883pc2(120)(10)(50/12)9approximate95%confidenceIntervalforProportionAttendingLocalUniversity:.702(.0883)1-444 DataandStatisticsor.5234to.876617(37)35(32)57(44)11,24017.a.x36.9737c173557304Estimateofmeanage:36.9737yearsb.ProportionofCollegeGraduates:128/304=.4211ProportionofMales:112/304=.36842c.()xxM=[17(37)-(36.9737)(17)]2+···+[57(44)-(36.9737)(44)]2ici=(.4471)2+(-174.0795)2+(-25.3162)2+(-460.2642)2+(173.1309)2+(180.3156)2+(-94.7376)2+(400.4991)2=474,650.681508474,650.68s2.2394xc2(150)(8)(40)7approximate95%confidenceIntervalforMeanAgeofAgents:36.97372(2.2394)or32.4949to41.4525d.2()apM=[3-.4211(17)]2+···+[25-.4211(57)]2ici=(-4.1587)2+(-.7385)2+(-2.9486)2+(10.2074)2+(-.1073)2+(-3.0532)2+(-.2128)2+(.9973)2=141.09891508141.0989s.0386pc2(150)(8)(40)7approximate95%confidenceIntervalforProportionofAgentsthatareCollegeGraduates:.42112(.0386)or.3439to.49831-445 Chapter12e.()apM=[4-.3684(17)]2+···+[26-.3684(57)]2ici=(-2.2628)2+(-.8940)2+(-2.5784)2+(3.6856)2+(-3.8412)2+(1.5792)2+(-.6832)2+(5.0012)2=68.8787150868.8787s.0270pc2(150)(8)(40)7approximate95%confidenceIntervalforProportionofAgentsthatareMale:.36842(.0270)or.3144to.422418.a.p=0.19(0.19)(0.81)s0.0206p363Approximate95%ConfidenceInterval:0.192(0.0206)or0.1488to0.2312b.p=0.31(0.31)(0.69)s0.0243p363Approximate95%ConfidenceInterval:0.312(0.0243)or0.2615to0.3585c.p=0.17(0.17)(0.83)s0.0197p3731-446 DataandStatisticsApproximate95%ConfidenceInterval:0.172(0.0197)or0.1306to0.2094d.Thelargeststandarderroriswhenp=.50.Atp=.50,weget(0.5)(0.5)s0.0262p363Multiplyingby2,wegetaboundofB=2(.0262)=0.0525Forasampleof363,then,theyknowthatintheworstcase(p=0.50),theboundwillbeapproximately5%.e.Ifthepollwasconductedbycallingpeopleathomeduringthedaythesampleresultswouldonlyberepresentativeofadultsnotworkingoutsidethehome.ItislikelythattheLouisHarrisorganizationtookprecautionsagainstthisandotherpossiblesourcesofbias.19.a.Assume(N-n)/N1p=.55(0.55)(0.45)s0.0222p504b.p=.31(0.31)(0.69)s0.0206p504c.Theestimateofthestandarderrorinpart(a)islargerbecausepiscloserto.50.d.Approximate95%Confidenceinterval:.552(.0222)or1-447 Chapter1.5056to.5944e.Approximate95%Confidenceinterval:.312(.0206).2688to.35123000200300020.a.s204.9390x3000200Approximate95%ConfidenceIntervalforMeanAnnualSalary:23,2002(204.9390)or$22,790to$23,610b.Nx=3000(23,200)=69,600,000s=3000(204.9390)=614,817xApproximate95%ConfidenceIntervalforPopulationTotalSalary:69,600,0002(614,817)or$68,370,366to$70,829,634c.p=.733000200(.73)(.27)s.0304p3000199Approximate95%ConfidenceIntervalforProportionthatareGenerallySatisfied:.732(.0304)or.6692to.7908d.Ifmanagementadministeredthequestionnaireandanonymitywasnotguaranteedwewouldexpectadefiniteupwardbiasinthepercentreportingtheywere“generallysatisfied”withtheirjob.Aprocedureforguaranteeinganonymityshouldreducethebias.21.a.p=1/31-448 DataandStatistics38030(1/3)(2/3)s.0840p38029Approximate95%ConfidenceInterval:.33332(.0840)or.1653to.5013b.X2=760(19/45)=320.8889c.p=19/45=.422276045(19/45)(26/45)s.0722p76044Approximate95%ConfidenceInterval:.42222(.0722)or.2778to.566638010760192607d.p.3717st140030140045140025pp(1)(1/3)(2/3)hhNNn()380(350)hhhn129h(19/45)(26/45)(7/25)(18/25)760(715)260(235)4424=1019.1571+3012.7901+513.2400=4545.18921s4545.1892.0482pst2(1400)Approximate95%ConfidenceInterval:.37172(.0482)or.2753to.46811-449 Chapter122.a.X=380(9/30)+760(12/45)+260(11/25)=431.0667Estimateapproximately431deathsduetobeating.38097601226011b.p.3079st140030140045140025pphh(1)NNn()hhhn1h=(380)(380-30)(9/30)(21/30)/29+(760)(760-45)(12/45)(33/45)/44+(260)(260-25)(11/25)(14/25)/24=4005.50791s4005.5079.0452pst2(1400)Approximate95%ConfidenceInterval:.30792(.0452)or.2175to.3983380217603426015c.p.7116st140030140045140025pphh(1)NNn()hhhn1h=(380)(380-30)(21/30)(9/30)/29+(760)(760-45)(34/45)(11/45)/44+(260)(260-25)(15/25)(10/25)/24=3855.04171s3855.0417.0443pst2(1400)Approximate95%ConfidenceInterval:.71162(.0443)1-450 DataandStatisticsor.6230to.8002d.X=1400(.7116)=996.24Estimateoftotalnumberofblackvictims99623000(80)600(150)250(220)100(700)50(3000)23.a.n222(20)2222(4000)3000(80)600(150)250(220)100(700)50(3000)4366,025,000,000170.73651,600,000,000543,800,000Roundingup,weneedasamplesizeof171forthedesiredprecision.3000(80)n171681605,000600(150)n171252605,000250(220)n171163605,000100(700)n171204605,00050(3000)n171425605,00014(61)7(74)96(78)23(69)71(73)29(84)18,06624.a.x75.275c14796237129240Estimateofmeanageisapproximately75yearsold.122308102284b.p.35c147962371292402()apM=[12-.35(14)]2+[2-.35(7)]2+[30-.35(96)]2ici+[8-.35(23)]2+[10-.35(71)]2+[22-.35(29)]2=(7.1)2+(-.45)2+(-3.6)2+(-.05)2+(-14.85)2+(11.85)2=424.521-451 Chapter11006424.52s.0760pc2(100)(6)(48)5Approximate95%ConfidenceInterval:.352(.0760)or.198to.502X=4800(.35)=1680EstimateoftotalnumberofDisabledPersonsis1680.1-452'