總體正態(tài)分布課件

ChapterSevenParametricTestsofSignificanceOutlineIntroductionTestingthesignificancebetweentwomeans1)Normaldistributionandpopulationvarianceknown;OutlineII.Testingthesignificancebetweentwomeans2)Normaldistributionandpopulationvarianceunknown;OutlineII.Testingthesignificancebetweentwomeans3)Non-normaldistribution.OutlineIII.Testingthesignificanceofdifferencesbetweentwomeans1)Testsforindependentsamples;a.Thez-testforlargesamplesb.Thet-testforsmallsamplesOutlineIII.Testingthesignificanceofdifferencesbetweentwomeans2)At-testforcorrelatedsamples;3)Decidingwhichtesttouse.OutlineIV.Testingthesignificanceofdifferencesbetweentwoproportions,forindependentsamples

IntroductionLet

usfirstremindourselvesoftheconditionsfortheapplicabilityofparametrictests.Thedatamusthaveatleastanintervallevelofmeasurement;Thetestsare,strictlyspeaking,applicableonlywherethepopulationsfromwhichthesamplesaretakenarenormal,thoughthisrequirementcanberelaxedinthecaseoflargesamples.Sometestsalsorequirethevariancesofthepopulationstobeapproximatelyequal.

TestingthesignificancebetweentwomeansItisakindoftestfortestingsignificancebetweenpopulationmeanandsamplemean.Normaldistributionandpopulationvarianceknown總體正態(tài)分布、總體方差已知

Inthiscase,wewillusez-testbycomparingtheabsolutevalueofstatistic

ZandcriticalvalueofZ.(Normaldistributiontable)Normaldistributionandpopulationvarianceunknown總體正態(tài)分布、總體方差未知

Inthiscase,wewilluset-testbycomparingtheabsolutevalueofstatistic

tandcriticalvalueoft.(tdistributiontable)Non-Normaldistribution總體非正態(tài)分布

Inthiscase,wewillnotsimplyusez-testort-test,especiallywhenthedistributionofthepopulationisnotnormal.Wecanusenon-parametrictestsinsteadorZ’test.TestingthesignificanceofdifferencesbetweentwomeansTestsforindependentsamples

獨(dú)立樣本檢驗(yàn)Thez-testforlargesamples（樣本大于30）

Bothpopulations’variancesknown,usez-test

Z=(x1-x2)-(μ1-μ2)/√σ21/n1+σ22/n2b.Thet-testforsmallsamples

Bothpopulations’variancesunknown,uset-testTestsforindependentsamples

獨(dú)立樣本檢驗(yàn)Forthiscase,therearetwoconditions:Bothpopulations’variancesareequal(orwecanjustsupposetheyareequalfromthevariancesofthesamples)RefertotheformulaonPage78ofthebluebook.Testsforindependentsamples

獨(dú)立樣本檢驗(yàn)Forthiscase,therearetwoconditions:2)Bothpopulations’variancesareunequal,useCochran-CoxTest.RefertotheformulaonPage80ofthebluebook.ExerciseGroup1(N1=10)Group2(N2=9)

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17At-testforcorrelatedsamples相關(guān)樣本t檢驗(yàn)Forthiscase,therearealsotwoconditions:1)Correlatedcoefficientunknown.RefertotheformulaonPage83ofthebluebook.t=Σd/√[NΣd2-(Σd)2]/N-1At-testforcorrelatedsamples相關(guān)樣本t檢驗(yàn)Forthiscase,therearealsotwoconditions:2)Correlatedcoefficientknown.RefertotheformulaonPage85ofthebluebook.DecidingwhichtesttouseStart

Arethedataatleastinterval?Arethesamplesindependent?Arethesampleslarge?Usez-testyesyesyesUseanon-parametrictest

Uset-testforcorrelatedsamples

Arepopulationsalmostnormalandofequalvariance?Uset-testforindependentsamplesyesnonononoTestingthesignificanceofdifferencesbetweentwoproportions,forindependentsamplesForinstance,wemighttakerandomsamplesofauxiliaryverbsfromtwoEnglishtexts,andfindthatinonesample28auxiliariesoutof100weremodalverbs,whileintheothersample34outof90weremodalverbs.Thequestiontobeanswerediswhetherthisdifferenceintheproportionofmodalsissignificant,atsomechosenlevel(say1percent)Wecantakethefollowingsteps:f1=28f2=34N1=100N2=90P1=28/100=0.28P2=34/90=0.38Pp=f1+f2/N1+N2=62/190=0.33Standarderrorofdifference=√Pp(1-Pp)(1/N1+1/N2)=0.07Then,Z=P1-P2/Standarderrorofdifference=0.28-0.38/0.07=-1.43Thevalueobtainedissmallerthanthecriticalvalueforthe1percentlevelinanon-directionaltest,whichis2.58.Wethereforeconcludethatweareunabletodemonstrateasignificantdifferenceintheproportionsatthe1percentlevel.Exercises

Thefollowingscoresareobtainedbytwogroupsofsubjectsonalanguageproficiencytest:GroupA:41,58,62,