总体正态分布课件

ChapterSevenParametricTestsofSignificanceChapterSeven1OutlineIntroductionTestingthesignificancebetweentwomeans1)Normaldistributionandpopulationvarianceknown;OutlineIntroduction2OutlineII.Testingthesignificancebetweentwomeans2)Normaldistributionandpopulationvarianceunknown;OutlineII.Testingthesignifi3OutlineII.Testingthesignificancebetweentwomeans3)Non-normaldistribution.OutlineII.Testingthesignifi4OutlineIII.Testingthesignificanceofdifferencesbetweentwomeans1)Testsforindependentsamples;a.Thez-testforlargesamplesb.Thet-testforsmallsamplesOutlineIII.Testingthesignif5OutlineIII.Testingthesignificanceofdifferencesbetweentwomeans2)At-testforcorrelatedsamples;3)Decidingwhichtesttouse.OutlineIII.Testingthesignif6OutlineIV.Testingthesignificanceofdifferencesbetweentwoproportions,forindependentsamples

OutlineIV.Testingthesignifi7IntroductionIntroduction8Let

usfirstremindourselvesoftheconditionsfortheapplicabilityofparametrictests.Thedatamusthaveatleastanintervallevelofmeasurement;Thetestsare,strictlyspeaking,applicableLetusfirstremindourselves9onlywherethepopulationsfromwhichthesamplesaretakenarenormal,thoughthisrequirementcanberelaxedinthecaseoflargesamples.Sometestsalsorequirethevariancesofthepopulationstobeapproximatelyequal.onlywherethepopulations10

TestingthesignificancebetweentwomeansTestingthesignificanceb11Itisakindoftestfortestingsignificancebetweenpopulationmeanandsamplemean.Itisakindoftestfortesti12Normaldistributionandpopulationvarianceknown总体正态分布、总体方差已知

Inthiscase,wewillusez-testbycomparingtheabsolutevalueofstatistic

ZandcriticalvalueofZ.(Normaldistributiontable)Normaldistributionandpopula13Normaldistributionandpopulationvarianceunknown总体正态分布、总体方差未知

Inthiscase,wewilluset-testbycomparingtheabsolutevalueofstatistic

tandcriticalvalueoft.(tdistributiontable)Normaldistributionandpopula14Non-Normaldistribution总体非正态分布

Inthiscase,wewillnotsimplyusez-testort-test,especiallywhenthedistributionofthepopulationisnotnormal.Wecanusenon-parametrictestsinsteadorZ’test.Non-Normaldistribution15TestingthesignificanceofdifferencesbetweentwomeansTestingthesignificanceofdi16Testsforindependentsamples

独立样本检验Thez-testforlargesamples（样本大于30）

Bothpopulations’variancesknown,usez-test

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

Bothpopulations’variancesunknown,uset-testTestsforindependentsamples17Testsforindependentsamples

独立样本检验Forthiscase,therearetwoconditions:Bothpopulations’variancesareequal(orwecanjustsupposetheyareequalfromthevariancesofthesamples)RefertotheformulaonPage78ofthebluebook.Testsforindependentsamples18Testsforindependentsamples

独立样本检验Forthiscase,therearetwoconditions:2)Bothpopulations’variancesareunequal,useCochran-CoxTest.RefertotheformulaonPage80ofthebluebook.Testsforindependentsamples19ExerciseGroup1(N1=10)Group2(N2=9)

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17ExerciseGroup1(N1=10)20At-testforcorrelatedsamples相关样本t检验Forthiscase,therearealsotwoconditions:1)Correlatedcoefficientunknown.RefertotheformulaonPage83ofthebluebook.t=Σd/√[NΣd²-(Σd)²]/N-1At-testforcorrelatedsample21At-testforcorrelatedsamples相关样本t检验Forthiscase,therearealsotwoconditions:2)Correlatedcoefficientknown.RefertotheformulaonPage85ofthebluebook.At-testforcorrelatedsample22DecidingwhichtesttouseStart

Arethedataatleastinterval?Arethesamplesindependent?Arethesampleslarge?Usez-testyesyesyesUseanon-parametrictest

Uset-testforcorrelatedsamples

Arepopulationsalmostnormalandofequalvariance?Uset-testforindependentsamplesyesnonononoDecidingwhichtesttouseStar23Testingthesignificanceofdifferencesbetweentwoproportions,forindependentsamplesTestingthesignificanceofdi24Forinstance,wemighttakerandomsamplesofauxiliaryverbsfromtwoEnglishtexts,andfindthatinonesample28auxiliariesoutof100weremodalverbs,whileintheothersample34outof90weremodalverbs.Thequestiontobeanswerediswhetherthisdifferenceintheproportionofmodalsissignificant,atsomechosenlevel(say1percent)Forinstance,wemighttakera25Wecantakethefollowingsteps: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)=0.07Wecantakethefollowingstep26Then,Z=P1-P2/Standarderrorofdifference=0.28-0.38/0.07=-1.43Thevalueobtainedissmallerthanthecriticalvalueforthe1percentlevelinanon-directionaltest,whichis2.58.Wethereforeconcludethatweareunabletodemonstrateasignificantdifferenceintheproportionsatthe1percentlevel.Then,27ExercisesExercises28

Thefollowingscoresareobtainedbytwogroupsofsubjectsonalanguageproficiencytest:GroupA:41,58,62,51,48,34,64,50,53,60,44GroupB:38,40,64,47,51,49,32,44,61Theinvestigatorpredictsthatthemeanswilldiffersignificantly.Carryoutatesttoassesssignificanceatthe5percentlevel,givingreasonsforselectingthetestyoudecidetouse.Thefollowingscoresare29ThankYou!ThankYou!30ChapterSevenParametricTestsofSignificanceChapterSeven31OutlineIntroductionTestingthesignificancebetweentwomeans1)Normaldistributionandpopulationvarianceknown;OutlineIntroduction32OutlineII.Testingthesignificancebetweentwomeans2)Normaldistributionandpopulationvarianceunknown;OutlineII.Testingthesignifi33OutlineII.Testingthesignificancebetweentwomeans3)Non-normaldistribution.OutlineII.Testingthesignifi34OutlineIII.Testingthesignificanceofdifferencesbetweentwomeans1)Testsforindependentsamples;a.Thez-testforlargesamplesb.Thet-testforsmallsamplesOutlineIII.Testingthesignif35OutlineIII.Testingthesignificanceofdifferencesbetweentwomeans2)At-testforcorrelatedsamples;3)Decidingwhichtesttouse.OutlineIII.Testingthesignif36OutlineIV.Testingthesignificanceofdifferencesbetweentwoproportions,forindependentsamples

OutlineIV.Testingthesignifi37IntroductionIntroduction38Let

usfirstremindourselvesoftheconditionsfortheapplicabilityofparametrictests.Thedatamusthaveatleastanintervallevelofmeasurement;Thetestsare,strictlyspeaking,applicableLetusfirstremindourselves39onlywherethepopulationsfromwhichthesamplesaretakenarenormal,thoughthisrequirementcanberelaxedinthecaseoflargesamples.Sometestsalsorequirethevariancesofthepopulationstobeapproximatelyequal.onlywherethepopulations40

TestingthesignificancebetweentwomeansTestingthesignificanceb41Itisakindoftestfortestingsignificancebetweenpopulationmeanandsamplemean.Itisakindoftestfortesti42Normaldistributionandpopulationvarianceknown总体正态分布、总体方差已知

Inthiscase,wewillusez-testbycomparingtheabsolutevalueofstatistic

ZandcriticalvalueofZ.(Normaldistributiontable)Normaldistributionandpopula43Normaldistributionandpopulationvarianceunknown总体正态分布、总体方差未知

Inthiscase,wewilluset-testbycomparingtheabsolutevalueofstatistic

tandcriticalvalueoft.(tdistributiontable)Normaldistributionandpopula44Non-Normaldistribution总体非正态分布

Inthiscase,wewillnotsimplyusez-testort-test,especiallywhenthedistributionofthepopulationisnotnormal.Wecanusenon-parametrictestsinsteadorZ’test.Non-Normaldistribution45TestingthesignificanceofdifferencesbetweentwomeansTestingthesignificanceofdi46Testsforindependentsamples

独立样本检验Thez-testforlargesamples（样本大于30）

Bothpopulations’variancesknown,usez-test

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

Bothpopulations’variancesunknown,uset-testTestsforindependentsamples47Testsforindependentsamples

独立样本检验Forthiscase,therearetwoconditions:Bothpopulations’variancesareequal(orwecanjustsupposetheyareequalfromthevariancesofthesamples)RefertotheformulaonPage78ofthebluebook.Testsforindependentsamples48Testsforindependentsamples

独立样本检验Forthiscase,therearetwoconditions:2)Bothpopulations’variancesareunequal,useCochran-CoxTest.RefertotheformulaonPage80ofthebluebook.Testsforindependentsamples49ExerciseGroup1(N1=10)Group2(N2=9)

18131514131217614118131017111675

17ExerciseGroup1(N1=10)50At-testforcorrelatedsamples相关样本t检验Forthiscase,therearealsotwoconditions:1)Correlatedcoefficientunknown.RefertotheformulaonPage83ofthebluebook.t=Σd/√[NΣd²-(Σd)²]/N-1At-testforcorrelatedsample51At-testforcorrelatedsamples相关样本t检验Forthiscase,therearealsotwoconditions:2)Correlatedcoefficientknown.RefertotheformulaonPage85ofthebluebook.At-testforcorrelatedsample52DecidingwhichtesttouseStart

Arethedataatleastinterval?Arethesamplesindependent?Arethesampleslarge?Usez-testyesyesyesUseanon-parametrictest

Uset-testforcorrelatedsamples

Arepopulationsalmostnormalandofequalvariance?Uset-testforindependentsamplesyesnonononoDecidingwhichtesttouseStar53Testingthesignificanceofdifferencesbetweentwoproportions,forindependentsamplesTestingthesignificanceofdi54Forinstance,wemighttakerandomsamplesofauxiliaryverbsfromtwoEnglishtexts,andfindthatinonesample28auxiliariesoutof100weremodalverbs,whileintheothersample34outof90weremodalverbs.Thequestiontobeansweredis