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### Transcript of 155S8.1-2 3 Basics of Hypothesis · PDF file155S8.1­2_3 Basics of Hypothesis Testing 1...

• 155S8.12_3BasicsofHypothesisTesting

1

March30,2011

Chapter8HypothesisTesting

Variance

MAT155StatisticalAnalysisDr.ClaudeMoore

CapeFearCommunityCollege

Definitions

MainObjective

ExamplesofHypothesesthatcanbeTested Genetics:TheGenetics&IVFInstituteclaimsthatitsXSORTmethodallowscouplestoincreasetheprobabilityofhavingababygirl.

Medicine:Medicalresearchersclaimthatwhenpeoplewithcoldsaretreatedwithechinacea,thetreatmenthasnoeffect.

• 155S8.12_3BasicsofHypothesisTesting

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March30,2011

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:,.

• 155S8.12_3BasicsofHypothesisTesting

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Ifyouareconductingastudyandwanttouseahypothesistesttosupportyourclaim,theclaimmustbewordedsothatitbecomesthealternativehypothesis.

Figure82

Example:Step1: Expressthegivenclaiminsymbolicform.Theclaimthatthemeanisatmost195lbisexpressedinsymbolicformas

195lb.

Step2: If195lbisfalse,then>195lbmustbetrue.

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

• 155S8.12_3BasicsofHypothesisTesting

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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:

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

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

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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