统计学Chap010

McGraw-Hill/IrwinCopyright©2009byTheMcGraw-HillCompanies,Inc.AllRightsReserved.StatisticalInferencesBasedonTwoSamplesChapter1010-2StatisticalInferencesBasedonTwoSamples10.1ComparingTwoPopulationMeansbyUsingIndependentSamples:VariancesKnown10.2ComparingTwoPopulationMeansbyUsingIndependentSamples:VariancesUnknown10.3PairedDifferenceExperiments10.4ComparingTwoPopulationProportionsbyUsingLarge,IndependentSamples10.5ComparingTwoPopulationVariancesbyUsingIndependentSamples10-3ComparingTwoPopulationMeansbyUsingIndependentSamples:VariancesKnown•Supposearandomsamplehasbeentakenfromeachoftwodifferentpopulations•Supposethatthepopulationsareindependentofeachother–Thentherandomsamplesareindependentofeachother•Thenthesamplingdistributionofthedifferenceinsamplemeansisnormallydistributed10-4SamplingDistributionoftheDifferenceofTwoSampleMeans#1•Supposepopulation1hasmeanµ1andvarianceσ12–Frompopulation1,arandomsampleofsizen1isselectedwhichhasmeanx1andvariances12•Supposepopulation2hasmeanµ2andvarianceσ22–Frompopulation2,arandomsampleofsizen2isselectedwhichhasmeanx2andvariances22•Thenthesampledistributionofthedifferenceoftwosamplemeans…10-5SamplingDistributionoftheDifferenceofTwoSampleMeans#2•Isnormal,ifeachofthesampledpopulationsisnormal–Approximatelynormalifthesamplesizesn1andn2arelarge•Hasmeanµx1–x2=µ1–µ2•Hasstandarddeviation22212121nnxx10-6SamplingDistributionoftheDifferenceofTwoSampleMeans#310-7z-BasedConfidenceIntervalfortheDifferenceinMeans(VariancesKnown)•A100(1–)percentconfidenceintervalforthedifferenceinpopulationsµ1–µ2is222121221nnzxx10-8z-BasedTestAbouttheDifferenceinMeans(VariancesKnown)•TestthenullhypothesisaboutH0:µ1–µ2=D0–D0=µ1–µ2istheclaimeddifferencebetweenthepopulationmeans–D0isanumberwhosevaluevariesdependingonthesituation–OftenD0=0,andthenullmeansthatthereisnodifferencebetweenthepopulationmeans10-9z-BasedTestAbouttheDifferenceinMeans(VariancesKnown)•Usethenotationfromtheconfidenceintervalstatementonapriorslide•Assumethateachsampledpopulationisnormalorthatthesamplessizesn1andn2arelarge10-10TestStatistic(VariancesKnown)•Theteststatisticis•Thesamplingdistributionofthisstatisticisastandardnormaldistribution•Ifthepopulationsarenormalandthesamplesareindependent...222121021nnDxxz10-11z-BasedTestAbouttheDifferenceinMeans(VariancesKnown)•RejectH0:µ1–µ2=D0infavorofaparticularalternativehypothesisatalevelofsignificanceiftheappropriaterejectionpointruleholdsorifthecorrespondingp-valueislessthan•Rulesareonthenextslide…10-12z-BasedTestAbouttheDifferenceinMeans(VariancesKnown)ContinuedAlternativeRejectH0ifp-valueHa:µ1–µ2D0zzAreaunderstandardnormaltotherightof+zHa:µ1–µ2D0zzAreaunderstandardnormaltotheleftof-zHa:µ1–µ2≠D0|z|z/2*Twicetheareunderstandardnormaltotherightof|z|*Eitherzz/2orz-z/210-13ComparingTwoPopulationMeansbyUsingIndependentSamples:VariancesUnknown•Generally,thetruevaluesofthepopulationvariancesσ12andσ22arenotknown•Theyhavetobeestimatedfromthesamplevariancess12ands22,respectively10-14ComparingTwoPopulationMeansbyUsingIndependentSamples:VariancesUnknown#2•Alsoneedtoestimatethestandarddeviationofthesamplingdistributionofthedifferencebetweensamplemeans•Twoapproaches:1.Ifitcanbeassumedthatσ12=σ22=σ2,thencalculatethe“pooledestimate”ofσ22.Ifσ12≠σ22,thenuseapproximatemethods10-15PooledEstimateofσ2•Assumethatσ12=σ22=σ2•Thepooledestimateofσ2istheweightedaveragesofthetwosamplevariances,s12ands22•Thepooledestimateofσ2isdenotedbysp2•Theestimateofthepopulationstandarddeviationofthesamplingdistributionis211212222112nnsnsnsp2121121nnspxx10-16t-BasedConfidenceIntervalfortheDifferenceinMeans(VariancesUnknown)•Selectindependentrandomsamplesfromtwonormalpopulationswithequalvariances•A100(1–)percentconfidenceintervalforthedifferenceinpopulationsµ1–µ2is•where•andt/2isbasedon(n1+n2-2)degreesoffreedom(df)21222111nnstxxp211212222112nnsnsnsp10-17TestStatistic(VariancesUnknown)•Theteststatisticis•whereD0=µ1–µ2istheclaimeddifferencebetweenthepopulationmeans•Thesamplingdistributionofthisstatisticisatdistributionwith(n1+n2–2)degreesoffreedom21202111nnsDxxp10-18t-BasedTestAbouttheDifferenceinMeans(VariancesUnknown)AlternativeRejectH0ifp-valueHa:µ1–µ2D0ttAreaunderstandardnormaltotherightof+tHa:µ1–µ2D0ttAreaunderstandardnormaltotheleftof-tHa:µ1–µ2≠D0|t|t/2*Twicetheareunderstandardnormaltotherightof|t|wheret,t/2,andp-valuesarebasedon(n1+n2-2)degreesoffreedom*Eithertt/2ort-t/210-19t-BasedConfidenceIntervalsandTestsforDifferenceswithUnequalVariances•Ifpopulationsarenormal,butsamplesizesandvariancesdiffersubstantially,small-sampleestimationandtestingcanbebasedonthese“unequalvariance”procedure•Confidenceinterval•Teststatistics222121/221nsnstxx222121021nsnsDxxt10-20t-BasedConfidenceIntervalsandTestsforDifferenceswithUnequalVariances#2•Forboththeconfidenceintervalandhypothesistest,thedegreesoffreedomareequalto…1122222121212222121n/nsn/ns/ns/nsdf10-21PairedDifferenceExperiments•Before,drewrandomsamples