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  • 155S8.12_3BasicsofHypothesisTesting

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    Chapter8HypothesisTesting

    81ReviewandPreview82BasicsofHypothesisTesting83TestingaClaimaboutaProportion84TestingaClaimAboutaMean:Known85TestingaClaimAboutaMean:NotKnown86TestingaClaimAboutaStandardDeviationor

    Variance

    MAT155StatisticalAnalysisDr.ClaudeMoore

    CapeFearCommunityCollege

    S4.D2.Chapter8HypothesisTesting155Chapter8Lesson(Packagefile)Thesenotescoverthefollowingtopics:BasicsofHypothesisTestingTestingClaimaboutProportionTestingClaimofMean,knownTestingClaimofMean,unknownTestingClaimaboutStandardDeviation

    ReviewInChapters2and3weuseddescriptivestatisticswhenwesummarizeddatausingtoolssuchasgraphs,andstatisticssuchasthemeanandstandarddeviation.Methodsofinferentialstatisticsusesampledatatomakeaninferenceorconclusionaboutapopulation.Thetwomainactivitiesofinferentialstatisticsareusingsampledatato(1)estimateapopulationparameter(suchasestimatingapopulationparameterwithaconfidenceinterval),and(2)testahypothesisorclaimaboutapopulationparameter.InChapter7wepresentedmethodsforestimatingapopulationparameterwithaconfidenceinterval,andinthischapterwepresentthemethodofhypothesistesting.

    Definitions

    Instatistics,ahypothesisisaclaimorstatementaboutapropertyofapopulation.

    Ahypothesistest(ortestofsignificance)isastandardprocedurefortestingaclaimaboutapropertyofapopulation.

    MainObjective

    Themainobjectiveofthischapteristodeveloptheabilitytoconducthypothesistestsforclaimsmadeaboutapopulationproportionp,apopulationmean,orapopulationstandarddeviation.

    ExamplesofHypothesesthatcanbeTested Genetics:TheGenetics&IVFInstituteclaimsthatitsXSORTmethodallowscouplestoincreasetheprobabilityofhavingababygirl.

    Business:Anewspaperheadlinemakestheclaimthatmostworkersgettheirjobsthroughnetworking.

    Medicine:Medicalresearchersclaimthatwhenpeoplewithcoldsaretreatedwithechinacea,thetreatmenthasnoeffect.

    AircraftSafety:TheFederalAviationAdministrationclaimsthatthemeanweightofanairlinepassenger(includingcarryonbaggage)isgreaterthan185lb,whichitwas20yearsago.

    QualityControl:Whennewequipmentisusedtomanufactureaircraftaltimeters,thenewaltimetersarebetterbecausethevariationintheerrorsisreducedsothatthereadingsaremoreconsistent.(Inmanyindustries,thequalityofgoodsandservicescanoftenbeimprovedbyreducingvariation.)

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    CautionWhenconductinghypothesistestsasdescribedinthischapterandthefollowingchapters,insteadofjumpingdirectlytoproceduresandcalculations,besuretoconsiderthecontextofthedata,thesourceofthedata,andthesamplingmethodusedtoobtainthesampledata.

    KeyConcept

    Thissectionpresentsindividualcomponentsofahypothesistest.Weshouldknowandunderstandthefollowing: Howtoidentifythenullhypothesisandalternativehypothesisfromagivenclaim,andhowtoexpressbothinsymbolicform Howtocalculatethevalueoftheteststatistic,givenaclaimandsampledata Howtoidentifythecriticalvalue(s),givenasignificancelevel HowtoidentifythePvalue,givenavalueoftheteststatistic Howtostatetheconclusionaboutaclaiminsimpleandnontechnicalterms

    Part1:TheBasicsofHypothesisTesting

    RareEventRuleforInferentialStatisticsIf,underagivenassumption,theprobability

    ofaparticularobservedeventisexceptionallysmall,weconcludethattheassumptionisprobablynotcorrect.

    NullHypothesis:H0 Thenullhypothesis(denotedbyH0)isastatementthatthevalueofapopulationparameter(suchasproportion,mean,orstandarddeviation)isequaltosomeclaimedvalue. Wetestthenullhypothesisdirectly. EitherrejectH0orfailtorejectH0.

    AlternativeHypothesis:H1 Thealternativehypothesis(denotedbyH1orHaorHA)isthestatementthattheparameterhasavaluethatsomehowdiffersfromthenullhypothesis. Thesymbolicformofthealternativehypothesismustuseoneofthesesymbols:,.

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    NoteaboutFormingYourOwnClaims(Hypotheses)

    Ifyouareconductingastudyandwanttouseahypothesistesttosupportyourclaim,theclaimmustbewordedsothatitbecomesthealternativehypothesis.

    NoteaboutIdentifyingH0andH1

    Figure82

    Example:Considertheclaimthatthemeanweightofairlinepassengers(includingcarryonbaggage)isatmost195lb(thecurrentvalueusedbytheFederalAviationAdministration).FollowthethreestepprocedureoutlinedinFigure82toidentifythenullhypothesisandthealternativehypothesis.

    Example:Step1: Expressthegivenclaiminsymbolicform.Theclaimthatthemeanisatmost195lbisexpressedinsymbolicformas

    195lb.

    Step2: If195lbisfalse,then>195lbmustbetrue.

    Step3: Ofthetwosymbolicexpressions195lband>195lb,weseethat>195lbdoesnotcontainequality,soweletthealternativehypothesisH1be>195lb.Also,thenullhypothesismustbeastatementthatthemeanequals195lb,soweletH0be=195lb.

    Notethattheoriginalclaimthatthemeanisatmost195lbisneitherthealternativehypothesisnorthenullhypothesis.(However,wewouldbeabletoaddresstheoriginalclaimuponcompletionofahypothesistest.)

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    Theteststatisticisavalueusedinmakingadecisionaboutthenullhypothesis,andisfoundbyconvertingthesamplestatistictoascorewiththeassumptionthatthenullhypothesisistrue.

    TestStatistic TestStatisticTeststatisticfor

    proportion

    Teststatisticforstandarddeviation

    Teststatisticformean

    Example:LetsagainconsidertheclaimthattheXSORTmethodofgenderselectionincreasesthelikelihoodofhavingababygirl.PreliminaryresultsfromatestoftheXSORTmethodofgenderselectioninvolved14coupleswhogavebirthto13girlsand1boy.Usethegivenclaimandthepreliminaryresultstocalculatethevalueoftheteststatistic.Usetheformatoftheteststatisticgivenabove,sothatanormaldistributionisusedtoapproximateabinomialdistribution.(Thereareotherexactmethodsthatdonotusethenormalapproximation.)

    Example:TheclaimthattheXSORTmethodofgenderselectionincreasesthelikelihoodofhavingababygirlresultsinthefollowingnullandalternativehypothesesH0:p=0.5andH1:p>0.5.Weworkundertheassumptionthatthenullhypothesisistruewithp=0.5.Thesampleproportionof13girlsin14birthsresultsin.Usingp=0.5,,andn=14,wefindthevalueoftheteststatisticasfollows:

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    Example:

    Weknowfrompreviouschaptersthatazscoreof3.21isunusual(becauseitisgreaterthan2).Itappearsthatinadditiontobeinggreaterthan0.5,thesampleproportionof13/14or0.929issignificantlygreaterthan0.5.

    Example:Thefigurebelowshowsthatthesampleproportionof0.929doesfallwithintherangeofvaluesconsideredtobesignificantbecausetheyaresofarabove0.5thattheyarenotlikelytooccurbychance(assumingthatthepopulationproportionisp=0.5).

    CriticalRegionThecriticalregion(orrejectionregion)isthesetofallvaluesoftheteststatisticthatcauseustorejectthenullhypothesis.Forexample,seetheredshadedregioninthefigurebelow.

    SignificanceLevelThesignificancelevel(denotedby)istheprobabilitythattheteststatisticwillfallinthecriticalregionwhenthenullhypothesisisactuallytrue.ThisisthesameintroducedinSection72.Commonchoicesforare0.05,0.01,and0.10.

    CriticalValueAcriticalvalueisanyvaluethatseparatesthecriticalregion(wherewerejectthenullhypothesis)fromthevaluesoftheteststatisticthatdonotleadtorejectionofthenullhypothesis.Thecriticalvaluesdependonthenatureofthenullhypothesis,thesamplingdistributionthatapplies,andthesignificancelevela.Seethepreviousfigurewherethecriticalvalueofz=1.645correspondstoasignificancelevelofa=0.05.

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    PValueThePvalue(orpvalueorprobabilityvalue)istheprobabilityofgettingavalueoftheteststatisticthatisatleastasextremeastheonerepresentingthesampledata,assumingthatthenullhypothesisistrue.

    Criticalregioninthelefttail:

    Pvalue=areatotheleftoftheteststatistic

    Criticalregionintherighttail:

    Pvalue=areatotherightoftheteststatistic

    Criticalregionintwotails:

    Pvalue=twicetheareainthetailbeyondtheteststatistic

    PValue

    ThenullhypothesisisrejectedifthePvalueisverysmall,suchas0.05orless.

    HereisamemorytoolusefulforinterpretingthePvalue:

    IfthePislow,thenullmustgo.IfthePishigh,thenullwillfly.

    ProcedureforFindingPValuesFigure85

    Caution

    DontconfuseaPvaluewithaproportionp.Knowthisdistinction:

    Pvalue= probabilityofgettingateststatisticatleastasextremeastheonerepresentingsampledata

    p=populationproportion

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    ExampleConsidertheclaimthatwiththeXSORTmethodofgenderselection,thelikelihoodofhavingababygirlisdifferentfromp=0.5,andusetheteststatisticz=3.21foundfrom13girlsin14births.

    Firstdeterminewhetherthegivenconditionsresultinacriticalregionintherighttail,lefttail,ortwotails,thenuseFigure85tofindthePvalue.InterpretthePvalue.

    ExampleTheclaimthatthelikelihoodofhavingababygirlisdifferentfromp=0.5canbeexpressedasp0.5sothecriticalregionisintwotails.UsingFigure85tofindthePvalueforatwotailedtest,weseethatthePvalueistwicetheareatotherightoftheteststatisticz=3.21.WerefertoTableA2(orusetechnology)tofindthattheareatotherightofz=3.21is0.0007.Inthiscase,thePvalueistwicetheareatotherightoftheteststatistic,sowehave:

    Pvalue=2(0.0007)=0.0014

    ExampleThePvalueis0.0014(or0.0013ifgreaterprecisionisusedforthecalculations).ThesmallPvalueof0.0014showsthatthereisaverysmallchanceofgettingthesampleresultsthatledtoateststatisticofz=3.21.ThissuggeststhatwiththeXSORTmethodofgenderselection,thelikelihoodofhavingababygirlisdifferentfrom0.5.

    TypesofHypothesisTests:Twotailed,Lefttailed,Righttailed

    Thetailsinadistributionaretheextremeregionsboundedbycriticalvalues.

    DeterminationsofPvaluesandcriticalvaluesareaffectedbywhetheracriticalregionisintwotails,thelefttail,ortherighttail.Itthereforebecomesimportanttocorrectlycharacterizeahypothesistestastwotailed,lefttailed,orrighttailed.

  • 155S8.12_3BasicsofHyp