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BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc.Chap10-1Chapter11One-WayANOVABusinessStatistics:AFirstCourse

FifthEditionBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc.Chap10-2LearningObjectivesInthischapter,youlearn:

Howtouseone-wayanalysisofvariance(ANOVA)totestfordifferencesamongthemeansofseveralpopulationsHowtoperformmultiplecomparisonsinaone-wayanalysisofvariance.BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc.Chap10-3GeneralANOVASettingInvestigatorcontrolsoneormorefactorsofinterestEachfactorcontainstwoormorelevelsLevelscanbenumericalorcategoricalDifferentlevelsproducedifferentgroupsThinkofeachgroupasasamplefromadifferentpopulationObserveeffectsonthedependentvariableArethegroupsthesame?Experimentaldesign:theplanusedtocollectthedataBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc.Chap10-4CompletelyRandomizedDesignExperimentalunits(subjects)areassignedrandomlytogroupsSubjectsareassumedhomogeneousOnlyonefactororindependentvariableWithtwoormorelevelsAnalyzedbyone-factoranalysisofvariance(ANOVA)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-5One-WayAnalysisofVarianceEvaluatethedifferenceamongthemeansofthreeormoregroupsExamples:Accidentratesfor1st,2nd,and3rdshiftExpectedmileageforfivebrandsoftiresAssumptionsPopulationsarenormallydistributedPopulationshaveequalvariancesSamplesarerandomlyandindependentlydrawnBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc.Chap10-6HypothesesofOne-WayANOVA

Allpopulationmeansareequali.e.,nofactoreffect(novariationinmeansamonggroups)

Atleastonepopulationmeanisdifferenti.e.,thereisafactoreffectDoesnotmeanthatallpopulationmeansaredifferent(somepairsmaybethesame)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-7One-WayANOVATheNullHypothesisisTrueAllMeansarethesame:(NoFactorEffect)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc.Chap10-8One-WayANOVATheNullHypothesisisNOTtrueAtleastoneofthemeansisdifferent(FactorEffectispresent)or(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-9PartitioningtheVariationTotalvariationcanbesplitintotwoparts:SST=TotalSumofSquares

(Totalvariation)SSA=SumofSquaresAmongGroups

(Among-groupvariation)SSW=SumofSquaresWithinGroups

(Within-groupvariation)SST=SSA+SSW组内变差组间变差总变差BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-10PartitioningtheVariationTotalVariation=theaggregatevariationoftheindividualdatavaluesacrossthevariousfactorlevels(SST)Within-GroupVariation=variationthatexistsamongthedatavalueswithinaparticularfactorlevel(SSW)Among-GroupVariation=variationamongthefactorsamplemeans(SSA)SST=SSA+SSW(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-11PartitionofTotalVariationVariationDuetoFactor(SSA)VariationDuetoRandomError(SSW)TotalVariation(SST)=+BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-12TotalSumofSquaresWhere:

SST=Totalsumofsquares

c=numberofgroupsorlevels

nj=numberofobservationsingroupj

Xij=ithobservationfromgroupj

X=grandmean(meanofalldatavalues)SST=SSA+SSWBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-13TotalVariation(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-14Among-GroupVariationWhere:

SSA=Sumofsquaresamonggroups

c=numberofgroups

nj=samplesizefromgroupj

Xj=samplemeanfromgroupj

X=grandmean(meanofalldatavalues)SST=SSA+SSWBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-15Among-GroupVariationVariationDuetoDifferencesAmongGroupsMeanSquareAmong=SSA/degreesoffreedom(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-16Among-GroupVariation(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-17Within-GroupVariationWhere:

SSW=Sumofsquareswithingroups

c=numberofgroups

nj=samplesizefromgroupj

Xj=samplemeanfromgroupj

Xij=ithobservationingroupjSST=SSA+SSWBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-18Within-GroupVariationSummingthevariationwithineachgroupandthenaddingoverallgroupsMeanSquareWithin=SSW/degreesoffreedom(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-19Within-GroupVariation(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-20ObtainingtheMeanSquaresTheMeanSquaresareobtainedbydividingthevarioussumofsquaresbytheirassociateddegreesoffreedomMeanSquareAmong(d.f.=c–1)MeanSquareWithin(d.f.=n–c)MeanSquareTotal(d.f.=n–1)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-21One-WayANOVATableSourceofVariationSumOfSquaresDegreesofFreedomMeanSquare(Variance)AmongGroupsc–1MSA=WithinGroupsSSWn–cMSW=TotalSSTn–1SSAMSAMSWFc=numberofgroupsn=sumofthesamplesizesfromallgroupsdf=degreesoffreedomSSAc

–

1SSWn

–

cFSTAT=BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-22One-WayANOVA

FTestStatisticTeststatistic

MSAismeansquaresamonggroups

MSWismeansquareswithingroupsDegreesoffreedomdf1=c–1(c=numberofgroups)df2=n–c(n=sumofsamplesizesfromallpopulations)H0:μ1=μ2=…

=μcH1:AtleasttwopopulationmeansaredifferentBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-23InterpretingOne-WayANOVA

FStatisticTheFstatisticistheratiooftheamongestimateofvarianceandthewithinestimateofvarianceTheratiomustalwaysbepositive

df1=c–1willtypicallybesmalldf2=n–cwilltypicallybelargeDecisionRule:RejectH0ifFSTAT>Fα,otherwisedonotrejectH00

RejectH0DonotrejectH0FαBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-24One-WayANOVA

FTestExampleYouwanttoseeifthreedifferentgolfclubsyielddifferentdistances.Yourandomlyselectfivemeasurementsfromtrialsonanautomateddrivingmachineforeachclub.Atthe0.05significancelevel,isthereadifferenceinmeandistance?

Club1

Club2

Club3

254 234 200

263 218 222

241 235 197

237 227 206

251 216 204BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-25•••••One-WayANOVAExample:ScatterPlot270260250240230220210200190••••••••••Distance

Club1

Club2

Club3

254 234 200

263 218 222

241 235 197

237 227 206

251 216 204Club123BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-26One-WayANOVAExampleComputations

Club1

Club2

Club3

254 234 200

263 218 222

241 235 197

237 227 206

251 216 204X1=249.2X2=226.0X3=205.8X=227.0n1=5n2=5n3=5n=15c=3SSA=5(249.2–227)2+5(226–227)2+5(205.8–227)2=4716.4SSW=(254–249.2)2+(263–249.2)2+…+(204–205.8)2=1119.6MSA=4716.4/(3-1)=2358.2MSW=1119.6/(15-3)=93.3BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-27FSTAT

=25.275One-WayANOVAExampleSolutionH0:μ1=μ2=μ3H1:μjnotallequal

=0.05df1=2df2=12TestStatistic:Decision:Conclusion:RejectH0at

=0.05Thereisevidencethatatleastoneμjdiffersfromtherest0

=.05Fα

=3.89RejectH0DonotrejectH0CriticalValue:Fα

=3.89One-WayANOVAExampleBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap10-28One-WayANOVAExample:ScatterDiagramBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap10-29

One-WayANOVAExample:ComputationsBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap10-30

SummaryTableBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap10-31One-WayANOVAExample:SolutionBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap10-32

BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-33TheTukey-KramerProcedureTellswhichpopulationmeansaresignificantlydifferente.g.:μ1=μ2

μ3DoneafterrejectionofequalmeansinANOVAAllowspairedcomparisonsCompareabsolutemeandifferenceswithcriticalrangexμ1=μ2μ3BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-34Tukey-KramerCriticalRangewhere: Qα= UpperTailCriticalValuefromStudentized RangeDistributionwithcandn-cdegrees offreedom(seeappendixE.8table) MSW=MeanSquareWithin njandnj’=Samplesizesfromgroupsjandj’BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-35TheTukey-KramerProcedure:Example1.Computeabsolutemeandifferences:

Club1

Club2

Club3

254 234 200

263 218 222

241 235 197

237 227 206

251 216 2042.FindtheQαvaluefromthetableinappendixE.8with c=3and(n–c)=(15–3)=12degreesoffreedom:BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-36TheTukey-KramerProcedure:Example5.Alloftheabsolutemeandifferencesaregreaterthancriticalrange.Thereforethereisasignificantdifferencebetweeneachpairofmeansat5%levelofsignificance.Thus,with95%confidencewecanconcludethatthemeandistanceforclub1isgreaterthanclub2and3,andclub2isgreaterthanclub3.

3.ComputeCriticalRange:4.Compare:(continued)BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-37ANOVAAssumptionsRandomnessandIndependenceSelectrandomsamplesfromthecgroups(orrandomlyassignthelevels)NormalityThesamplevaluesforeachgrouparefromanormalpopulationHomogeneityofVarianceAllpopulationssampledfromhavethesamevarianceCanbetestedwithLevene’sTestBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-38ANOVAAssumptions

Levene’sTestTeststheassumptionthatthevariancesofeachpopulationareequal.First,definethenullandalternativehypotheses:H0:σ21=σ22=…=σ2cH1:Notallσ2jareequalSecond,computetheabsolutevalueofthedifferencebetweeneachvalueandthemedianofeachgroup.Third,performaone-wayANOVAontheseabsolutedifferences.BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-39LeveneHomogeneityOfVarianceTestExampleCalculateMediansClub1Club2Club3237216197241218200251227204Median254234206263235222CalculateAbsoluteDifferencesClub1Club2Club314117109400037212818H0:σ21=σ22=σ23H1:Notallσ2jareequalBusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc..Chap10-40LeveneHomogeneityOfVarianceTestExample(continued)Anova:SingleFactorSUMMARYGroupsCountSumAverageVarianceClub15397.836.2Club2535717.5Club35316.250.2SourceofVariationSSdfMSFP-valueFcritBetweenGroups6.423.20.0920.9123.885WithinGroups415.61234.6Total42214

Sincethep-valueisgreaterthan0.05wefailtorejectH0&concludethevariancesareequal.

BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc.Chap10-41e.g.Consideranexperimentwithfourgroups,witheightvaluesineach.FortheANOVAsummarytablebelow,fillinallthemissingresults:SourceDegreesofFreedomSumofSquaresSumofSquaresFAmonggroups80Withingroups

560Total

BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc.Chap10-42a.Atthe0.05levelofsignificance,statethedecisionrulefortestingthenullhypothesisthatallfourgroupshaveequalpopulationmeans.b.Atthe0.05levelofsignificance,whatistheupper­tailcriticalvaluefromtheStudentizedrangedistribution?c.ToperformtheTukey­Kramerprocedure,whatisthecriticalrange?BusinessStatistics:AFirstCourse,5e©2009Prentice-Hall,Inc.Chap10-43Amanufacturerofpenshashiredanadvertisingagencytodevelopanadvertisingcampaignfortheupcom­ingholidayseason.Toprepareforthisproject,theresearchdirectordecidestoinitiateastudyoftheeffectofadver­tisingonproductperception.Anexperimentisdesignedtocomparefivedifferentadvertisements.AdvertisementAgreatlyundersellsthepen’scharacteristics.Advertise­mentBslightlyundersellsthepen’scharacteristics.Ad­vertisementCslightlyoversellsthepen’scharacteristics.AdvertisementDgreatlyoversellsthepen’scharacteristics.AdvertisementEattemptstocorrectlystatethepen’schar­acteristics.Asampleof30adultrespondents,takenfromalargerfocusgroup,israndomlyassignedtothefiveadvertisements(sothatthereare6respondentstoeachadver­tisement).Afterreadingtheadvertisementanddevelopingasenseof“productexpectation,”allrespondentsunknow­inglyreceivethesamep

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