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1、Chap10-1Chapter 10Two-SampleTestsandOne-WayANOVAStatisticsForManagersUsingMicrosoftExcel6thEditionChap10-2LearningObjectivesIn this chapter, you learn hypothesis testing procedures to test:ThemeansoftwoindependentpopulationsThemeansoftworelatedpopulationsTheproportionsoftwoindependentpopulationsThev
2、ariancesoftwoindependentpopulationsThemeansofmorethantwopopulationsChap10-3ChapterOverviewOne-WayAnalysisofVariance(ANOVA)F-testTukey-KramertestTwo-SampleTestsPopulationMeans,IndependentSamplesMeans,RelatedSamplesPopulationProportionsPopulationVariancesChap10-4Two-SampleTestsTwo-SampleTestsPopulatio
3、nMeans,IndependentSamplesMeans,RelatedSamplesPopulationVariancesMean1vs.independentMean2Samepopulationbeforevs.aftertreatmentVariance1vs.Variance2Examples:PopulationProportionsProportion1vs.Proportion2Chap10-5DifferenceBetweenTwoMeansPopulationmeans,independentsamples1and2knownGoal:Testhypothesisorf
4、ormaconfidenceintervalforthedifferencebetweentwopopulationmeans,12ThepointestimateforthedifferenceisX1X2*1and2unknown,assumedequal1and2unknown,notassumedequalChap10-6IndependentSamplesPopulationmeans,independentsamplesDifferentdatasourcesUnrelatedIndependentSampleselectedfromonepopulationhasnoeffect
5、onthesampleselectedfromtheotherpopulationUsethedifferencebetween2samplemeansUseZtest,apooled-variancettest,oraseparate-variancettest*1and2known1and2unknown,assumedequal1and2unknown,notassumedequalChap10-7DifferenceBetweenTwoMeansPopulationmeans,independentsamples1and2known*UseaZ teststatisticUseSpto
6、estimateunknown,useatteststatisticandpooledstandarddeviation1and2unknown,assumedequal1and2unknown,notassumedequalUseS1andS2toestimateunknown1and2,useaseparate-variancettestChap10-8Populationmeans,independentsamples1and2known1and2KnownAssumptions:SamplesarerandomlyandindependentlydrawnPopulationdistr
7、ibutionsarenormalorbothsamplesizesare30Populationstandarddeviationsareknown*1and2unknown,assumedequal1and2unknown,notassumedequalChap10-9Populationmeans,independentsamples1and2knownandthestandarderrorofX1X2isWhen1and2areknownandbothpopulationsarenormalorbothsamplesizesareatleast30,theteststatisticis
8、aZ-value(continued)1and2Known*1and2unknown,assumedequal1and2unknown,notassumedequalChap10-10Populationmeans,independentsamples1and2knownTheteststatisticfor12is:1and2Known*(continued)1and2unknown,assumedequal1and2unknown,notassumedequalChap10-11HypothesisTestsforTwoPopulationMeansLower-tailtest:H0:1
9、2H1:12i.e.,H0:12 0H1:122i.e.,H0:120H1:120Two-tailtest:H0:1=2H1:12i.e.,H0:12=0H1:120TwoPopulationMeans,IndependentSamplesChap10-12TwoPopulationMeans,IndependentSamplesLower-tailtest:H0:12 0H1:120Two-tailtest:H0:12=0H1:120 /2 /2 -z-z/2zz/2RejectH0ifZZRejectH0ifZZ/2Hypothesistestsfor12Chap10-13Populati
10、onmeans,independentsamples1and2knownTheconfidenceintervalfor12is:ConfidenceInterval,1and2Known*1and2unknown,assumedequal1and2unknown,notassumedequalChap10-14Populationmeans,independentsamples1and2known1and2Unknown,AssumedEqualAssumptions:SamplesarerandomlyandindependentlydrawnPopulationsarenormallyd
11、istributedorbothsamplesizesareatleast30Populationvariancesareunknownbutassumedequal*1and2unknown,assumedequal1and2unknown,notassumedequalChap10-15Populationmeans,independentsamples1and2known(continued)*Formingintervalestimates:Thepopulationvariancesareassumedequal,sousethetwosamplevariancesandpoolth
12、emtoestimatethecommon2theteststatisticisatvaluewith(n1+n22)degreesoffreedom1and2unknown,assumedequal1and2unknown,notassumedequal1and2Unknown,AssumedEqualChap10-16Populationmeans,independentsamples1and2knownThepooledvarianceis(continued)*1and2unknown,assumedequal1and2unknown,notassumedequal1and2Unkno
13、wn,AssumedEqualChap10-17Populationmeans,independentsamples1and2knownWherethas(n1+n22)d.f.,andTheteststatisticfor12is:*(continued)1and2unknown,assumedequal1and2unknown,notassumedequal1and2Unknown,AssumedEqualChap10-18Populationmeans,independentsamples1and2knownTheconfidenceintervalfor12is:Where*Confi
14、denceInterval,1and2Unknown1and2unknown,assumedequal1and2unknown,notassumedequalChap10-19Pooled-VariancetTest:ExampleYouareafinancialanalystforabrokeragefirm.IsthereadifferenceindividendyieldbetweenstockslistedontheNYSE&NASDAQ?Youcollectthefollowingdata: NYSE NASDAQNumber 21 25Sample mean 3.27 2.53Sa
15、mple std dev 1.30 1.16Assumingbothpopulationsareapproximatelynormalwithequalvariances,isthereadifferenceinaverageyield(=0.05)?Chap10-20CalculatingtheTestStatisticTheteststatisticis:Chap10-21SolutionH0: 1 - 2 = 0 i.e. (1 = 2)H1: 1 - 2 0 i.e. (1 2) = 0.05df = 21 + 25 - 2 = 44Critical Values: t = 2.015
16、4Test Statistic:Decision:Conclusion:RejectH0at =0.05Thereisevidenceofadifferenceinmeans.t02.0154-2.0154.025RejectH0RejectH0.0252.040Chap10-22Populationmeans,independentsamples1and2known1and2Unknown,NotAssumedEqualAssumptions:SamplesarerandomlyandindependentlydrawnPopulationsarenormallydistributedorb
17、othsamplesizesareatleast30Populationvariancesareunknownbutcannotbeassumedtobeequal*1and2unknown,assumedequal1and2unknown,notassumedequalChap10-23Populationmeans,independentsamples1and2known(continued)*Formingtheteststatistic:Thepopulationvariancesarenotassumedequal,soincludethetwosamplevariancesinth
18、ecomputationofthet-teststatistictheteststatisticisatvalue(statisticalsoftwareisgenerallyusedtodothenecessarycomputations)1and2unknown,assumedequal1and2unknown,notassumedequal1and2Unknown,NotAssumedEqualChap10-24Populationmeans,independentsamples1and2knownTheteststatisticfor12is:*(continued)1and2unkn
19、own,assumedequal1and2unknown,notassumedequal1and2Unknown,NotAssumedEqualChap10-25RelatedPopulationsTestsMeansof2RelatedPopulationsPairedormatchedsamplesRepeatedmeasures(before/after)Usedifferencebetweenpairedvalues:EliminatesVariationAmongSubjectsAssumptions:BothPopulationsAreNormallyDistributedOr,i
20、fnotNormal,uselargesamplesRelatedsamplesDi = X1i - X2iChap10-26MeanDifference,DKnownTheithpaireddifferenceisDi,whereRelatedsamplesDi=X1i-X2iThepointestimateforthepopulationmeanpaireddifferenceisD:Supposethepopulationstandarddeviationofthedifferencescores,D,isknownnisthenumberofpairsinthepairedsample
21、Chap10-27TheteststatisticforthemeandifferenceisaZvalue:PairedsamplesMeanDifference,DKnown(continued)WhereD=hypothesizedmeandifferenceD=populationstandarddev.ofdifferencesn=thesamplesize(numberofpairs)Chap10-28ConfidenceInterval,DKnownTheconfidenceintervalforDisPairedsamplesWheren=thesamplesize(numbe
22、rofpairsinthepairedsample)Chap10-29IfDisunknown,wecanestimatetheunknownpopulationstandarddeviationwithasamplestandarddeviation:RelatedsamplesThesamplestandarddeviationisMeanDifference,DUnknownChap10-30Useapairedttest,theteststatisticforDisnowatstatistic,withn-1d.f.:PairedsamplesWherethasn-1d.f.andSD
23、is:MeanDifference,DUnknown(continued)Chap10-31TheconfidenceintervalforDisPairedsampleswhereConfidenceInterval,DUnknownChap10-32Lower-tailtest:H0:D 0H1:D0Two-tailtest:H0:D=0H1:D0PairedSamplesHypothesisTestingforMeanDifference,DUnknown /2 /2 -t-t/2tt/2RejectH0ifttRejectH0iftt/2Wherethasn-1d.f.Chap10-3
24、3Assumeyousendyoursalespeopletoa“customerservice”trainingworkshop.Hasthetrainingmadeadifferenceinthenumberofcomplaints?Youcollectthefollowingdata:PairedtTestExample Number of Complaints: (2) - (1)Salesperson Before (1) After (2) Difference, Di C.B. 6 4 - 2 T.F. 20 6 -14 M.H. 3 2 - 1 R.K. 0 0 0 M.O.
25、4 0 - 4 -21D=Din=-4.2Chap10-34Hasthetrainingmadeadifferenceinthenumberofcomplaints(atthe0.01level)? - 4.2D=H0: D = 0H1: D 0Test Statistic:Critical Value = 4.604 d.f. = n - 1 = 4Reject /2 - 4.604 4.604Decision: Do not reject H0(t stat is not in the reject region)Conclusion: There is not a significant
26、 change in the number of complaints.PairedtTest:SolutionReject /2 - 1.66 =.01Chap10-35TwoPopulationProportionsGoal:testahypothesisorformaconfidenceintervalforthedifferencebetweentwopopulationproportions,12ThepointestimateforthedifferenceisPopulationproportionsAssumptions:n115,n1(1-1)5n225,n2(1-2)5Ch
27、ap10-36TwoPopulationProportionsPopulationproportionsThepooledestimatefortheoverallproportionis:whereX1andX2arethenumbersfromsamples1and2withthecharacteristicofinterestSincewebeginbyassumingthenullhypothesisistrue,weassume1=2andpoolthetwosampleestimatesChap10-37TwoPopulationProportionsPopulationpropo
28、rtionsTheteststatisticforp1p2isaZstatistic:(continued)whereChap10-38ConfidenceIntervalforTwoPopulationProportionsPopulationproportionsTheconfidenceintervalfor12is:Chap10-39HypothesisTestsforTwoPopulationProportionsPopulationproportionsLower-tailtest:H0:1 2H1:12i.e.,H0:12 0H1:122i.e.,H0:120H1:120Two-
29、tailtest:H0:1=2H1:12i.e.,H0:12=0H1:120Chap10-40HypothesisTestsforTwoPopulationProportionsPopulationproportionsLower-tailtest:H0:12 0H1:120Two-tailtest:H0:12=0H1:120 /2 /2 -z-z/2zz/2RejectH0ifZZRejectH0ifZZ/2(continued)Chap10-41Example:TwopopulationProportionsIsthereasignificantdifferencebetweenthepr
30、oportionofmenandtheproportionofwomenwhowillvoteYesonPropositionA?Inarandomsample,36of72menand31of50womenindicatedtheywouldvoteYesTestatthe.05levelofsignificanceChap10-42Thehypothesistestis:H0:12=0(thetwoproportionsareequal)H1:120(thereisasignificantdifferencebetweenproportions)Thesampleproportionsar
31、e:Men:p1=36/72=.50Women:p2=31/50=.62Thepooledestimatefortheoverallproportionis:Example:TwopopulationProportions(continued)Chap10-43Theteststatisticfor12is:Example:TwopopulationProportions(continued).025-1.961.96.025-1.31Decision: Do not reject H0Conclusion: There is not significant evidence of a dif
32、ference in proportions who will vote yes between men and women.RejectH0RejectH0Critical Values = 1.96For = .05Chap10-44HypothesisTestsforVariancesTestsforTwoPopulationVariancesFteststatisticH0:12=22H1:1222Two-tailtestLower-tailtestUpper-tailtestH0:1222H1:1222*Chap10-45HypothesisTestsforVariancesTest
33、sforTwoPopulationVariancesFteststatisticTheFteststatisticis:=VarianceofSample1n1-1=numeratordegreesoffreedomn2-1=denominatordegreesoffreedom=VarianceofSample2*(continued)Chap10-46TheFcriticalvalue isfoundfromtheFtableTherearetwoappropriatedegreesoffreedom:numeratoranddenominatorIntheFtable,numerator
34、degreesoffreedomdeterminethecolumndenominatordegreesoffreedomdeterminetherowTheFDistributionwheredf1=n11;df2=n21Chap10-47F 0 FindingtheRejectionRegionnrejectionregionforatwo-tailtestis:FL RejectH0DonotrejectH0F 0 FU RejectH0DonotrejectH0F 0 /2RejectH0DonotrejectH0FU H0:12=22H1:1222H0:1222H1:1222FL /
35、2RejectH0RejectH0ifFFUChap10-48FindingtheRejectionRegionF 0 /2RejectH0DonotrejectH0FU H0:12=22H1:1222FL /2RejectH0(continued)2.FindFLusingtheformula:WhereFU*isfromtheFtablewithn21numeratorandn11denominatordegreesoffreedom(i.e.,switchthed.f.fromFU)1.FindFUfromtheFtableforn11numeratorandn21denominator
36、degreesoffreedomTofindthecriticalFvalues:Chap10-49FTest:AnExampleYouareafinancialanalystforabrokeragefirm.YouwanttocomparedividendyieldsbetweenstockslistedontheNYSE&NASDAQ.Youcollectthefollowingdata: NYSE NASDAQNumber 2125Mean3.272.53Std dev1.301.16IsthereadifferenceinthevariancesbetweentheNYSE&NASD
37、AQatthe = 0.05level?Chap10-50FTest:ExampleSolutionFormthehypothesistest:H0:2122=0(thereisnodifferencebetweenvariances)H1:21220(thereisadifferencebetweenvariances)Numerator:n11=211=20d.f.Denominator:n21=251=24d.f.FU=F.025,20,24=2.33FindtheFcriticalvaluesfor =0.05:Numerator:n21=251=24d.f.Denominator:n
38、11=211=20d.f.FL=1/F.025,24,20=1/2.41=0.415FU:FL:Chap10-51Theteststatisticis:0 /2=.025FU=2.33RejectH0DonotrejectH0H0:12=22H1:1222FTest:ExampleSolutionF=1.256isnotintherejectionregion,sowedonotrejectH0(continued)Conclusion:Thereisnotsufficientevidenceofadifferenceinvariancesat=.05FL=0.43/2=.025RejectH
39、0F Chap10-52Two-SampleTestsinEXCELForindependentsamples:IndependentsampleZtestwithvariancesknown:Tools|dataanalysis|z-test:twosampleformeansPooledvariancettest:Tools|dataanalysis|t-test:twosampleassumingequalvariancesSeparate-variancettest:Tools|dataanalysis|t-test:twosampleassumingunequalvariancesF
40、orpairedsamples(ttest):Tools|dataanalysis|t-test:pairedtwosampleformeansForvariances:Ftestfortwovariances:Tools|dataanalysis|F-test:twosampleforvariancesChap10-53One-WayAnalysisofVarianceOne-WayAnalysisofVariance(ANOVA)F-testTukey-KramertestChap10-54GeneralANOVASettingInvestigatorcontrolsoneormorein
41、dependentvariablesCalledfactors(ortreatmentvariables)Eachfactorcontainstwoormorelevels(orgroupsorcategories/classifications)ObserveeffectsonthedependentvariableResponsetolevelsofindependentvariableExperimentaldesign:theplanusedtocollectthedataChap10-55One-WayAnalysisofVarianceEvaluatethedifferenceam
42、ongthemeansofthreeormoregroupsExamples:Accidentratesfor1st,2nd,and3rdshiftExpectedmileageforfivebrandsoftiresAssumptionsPopulationsarenormallydistributedPopulationshaveequalvariancesSamplesarerandomlyandindependentlydrawnChap10-56HypothesesofOne-WayANOVAAllpopulationmeansareequali.e.,notreatmenteffe
43、ct(novariationinmeansamonggroups)Atleastonepopulationmeanisdifferenti.e.,thereisatreatmenteffectDoesnotmeanthatallpopulationmeansaredifferent(somepairsmaybethesame)Chap10-57One-WayANOVAAllMeansarethesame:TheNullHypothesisisTrue(NoTreatmentEffect)Chap10-58One-WayANOVAAtleastonemeanisdifferent:TheNull
44、HypothesisisNOTtrue(TreatmentEffectispresent)or(continued)Chap10-59PartitioningtheVariationTotalvariationcanbesplitintotwoparts:SST=TotalSumofSquares(Totalvariation)SSA=SumofSquaresAmongGroups(Among-groupvariation)SSW=SumofSquaresWithinGroups(Within-groupvariation)SST=SSA+SSWChap10-60Partitioningthe
45、VariationTotalVariation=theaggregatedispersionoftheindividualdatavaluesacrossthevariousfactorlevels(SST)Within-GroupVariation=dispersionthatexistsamongthedatavalueswithinaparticularfactorlevel(SSW)Among-GroupVariation=dispersionbetweenthefactorsamplemeans(SSA)SST=SSA+SSW(continued)Chap10-61Partition
46、ofTotalVariationVariation Due to Factor (SSA)Variation Due to Random Sampling (SSW)Total Variation (SST)Commonlyreferredtoas:SumofSquaresWithinSumofSquaresErrorSumofSquaresUnexplainedWithin-GroupVariationCommonlyreferredtoas:SumofSquaresBetweenSumofSquaresAmongSumofSquaresExplainedAmongGroupsVariati
47、on=+d.f.=n1d.f.=c1d.f.=ncChap10-62TotalSumofSquaresWhere:SST=Totalsumofsquaresc=numberofgroups(levelsortreatments)nj=numberofobservationsingroupjXij=ithobservationfromgroupjX=grandmean(meanofalldatavalues)SST=SSA+SSWChap10-63TotalVariation(continued)Chap10-64Among-GroupVariationWhere:SSA=Sumofsquare
48、samonggroupsc=numberofgroupsnj=samplesizefromgroupjXj=samplemeanfromgroupjX=grandmean(meanofalldatavalues)SST=SSA+SSWChap10-65Among-GroupVariationVariationDuetoDifferencesAmongGroupsMeanSquareAmong=SSA/degreesoffreedom(continued)Chap10-66Among-GroupVariation(continued)Chap10-67Within-GroupVariationW
49、here:SSW=Sumofsquareswithingroupsc=numberofgroupsnj=samplesizefromgroupjXj=samplemeanfromgroupjXij=ithobservationingroupjSST=SSA+SSWChap10-68Within-GroupVariationSummingthevariationwithineachgroupandthenaddingoverallgroupsMeanSquareWithin=SSW/degreesoffreedom(continued)Chap10-69Within-GroupVariation
50、(continued)Chap10-70ObtainingtheMeanSquaresChap10-71One-WayANOVATableSource of VariationdfSSMS(Variance)Among GroupsSSAMSA =Within Groupsn - cSSWMSW =Totaln - 1SST =SSA+SSWc - 1MSAMSWF ratioc=numberofgroupsn=sumofthesamplesizesfromallgroupsdf=degreesoffreedomSSAc - 1SSWn - cF =Chap10-72One-WayANOVAF
51、TestStatisticTeststatisticMSAismeansquaresamonggroupsMSWismeansquareswithingroupsDegreesoffreedomdf1=c1(c=numberofgroups)df2=nc(n=sumofsamplesizesfromallpopulations)H0:1=2=cH1:AtleasttwopopulationmeansaredifferentChap10-73InterpretingOne-WayANOVAFStatisticTheFstatisticistheratiooftheamongestimateofv
52、arianceandthewithinestimateofvarianceTheratiomustalwaysbepositive df1=c-1willtypicallybesmalldf2=n-cwilltypicallybelargeDecisionRule:RejectH0ifFFU,otherwisedonotrejectH00 =.05RejectH0DonotrejectH0FU Chap10-74One-WayANOVAFTestExampleYouwanttoseeifthreedifferentgolfclubsyielddifferentdistances.Yourand
53、omlyselectfivemeasurementsfromtrialsonanautomateddrivingmachineforeachclub.Atthe0.05significancelevel,isthereadifferenceinmeandistance?Club 1Club 2 Club 3254234 200263218 222241235 197237227 206251216 204Chap10-75One-WayANOVAExample:ScatterDiagram270260250240230220210200190DistanceClub 1Club 2 Club
54、3254234 200263218 222241235 197237227 206251216 204Club123Chap10-76One-WayANOVAExampleComputationsClub 1Club 2 Club 3254234 200263218 222241235 197237227 206251216 204X1=249.2X2=226.0X3=205.8X=227.0n1=5n2=5n3=5n=15c=3SSA=5(249.2227)2+5(226227)2+5(205.8227)2=4716.4SSW=(254249.2)2+(263249.2)2+(204205.
55、8)2=1119.6MSA=4716.4/(3-1)=2358.2MSW=1119.6/(15-3)=93.3Chap10-77F = 25.275One-WayANOVAExampleSolutionH0:1=2=3H1:jnotallequal=0.05df1=2df2=12Test Statistic: Decision:Conclusion:RejectH0at =0.05Thereisevidencethatatleastonej differsfromtherest0 =.05FU = 3.89RejectH0DonotrejectH0Critical Value: FU = 3.
56、89Chap10-78SUMMARYGroupsCountSumAverageVarianceClub151246249.2108.2Club25113022677.5Club351029205.894.2ANOVASource of VariationSSdfMSFP-valueF critBetweenGroups4716.422358.225.2754.99E-053.89WithinGroups1119.61293.3Total5836.014One-WayANOVAExcelOutputEXCEL:tools|dataanalysis|ANOVA:singlefactorChap10
57、-79TheTukey-KramerProcedureTellswhichpopulationmeansaresignificantlydifferente.g.:1=23DoneafterrejectionofequalmeansinANOVAAllowspair-wisecomparisonsCompareabsolutemeandifferenceswithcriticalrangex1=23Chap10-80Tukey-KramerCriticalRangewhere:QU=ValuefromStudentizedRangeDistributionwithcandn-cdegreeso
58、ffreedomforthedesiredlevelof(seeappendixE.7table)MSW=MeanSquareWithinnjandnj=SamplesizesfromgroupsjandjChap10-81TheTukey-KramerProcedure:Example1.Computeabsolutemeandifferences:Club 1Club 2 Club 3254234 200263218 222241235 197237227 206251216 2042.FindtheQUvaluefromthetableinappendixE.7withc=3and(nc
59、)=(153)=12degreesoffreedomforthedesiredlevelof(=0.05usedhere):Chap10-82TheTukey-KramerProcedure:Example5.Alloftheabsolutemeandifferencesaregreaterthancriticalrange.Thereforethereisasignificantdifferencebetweeneachpairofmeansat5%levelofsignificance.Thus,with95%confidencewecanconcludethatthemeandistan
60、ceforclub1isgreaterthanclub2and3,andclub2isgreaterthanclub3. 3.ComputeCriticalRange:4.Compare:(continued)Chap10-83ChapterSummaryComparedtwoindependentsamplesPerformedZtestforthedifferenceintwomeansPerformedpooledvariancettestforthedifferenceintwomeansPerformedseparate-variancettestfordifferenceintwo
61、meansFormedconfidenceintervalsforthedifferencebetweentwomeansComparedtworelatedsamples(pairedsamples)PerformedpairedsampleZandttestsforthemeandifferenceFormedconfidenceintervalsforthemeandifferenceChap10-84ChapterSummaryComparedtwopopulationproportionsFormedconfidenceintervalsforthedifferencebetween
62、twopopulationproportionsPerformedZ-testfortwopopulationproportionsPerformedFtestsforthedifferencebetweentwopopulationvariancesUsedtheFtabletofindFcriticalvaluesDescribedone-wayanalysisofvarianceThelogicofANOVAANOVAassumptionsFtestfordifferenceincmeansTheTukey-Kramerprocedureformultiplecomparisons(continued)
