HypothesisTesting統(tǒng)計(jì)學(xué)假設(shè)檢驗(yàn)課件

HypothesisTesting統(tǒng)計(jì)學(xué)假設(shè)檢驗(yàn)1HypothesisTesting統(tǒng)計(jì)學(xué)假設(shè)檢驗(yàn)1HypothesisTesting9.1 NullandAlternativeHypothesesandErrorsinTesting9.2 zTestsaboutaPopulationwithknowns9.3 tTestsaboutaPopulationwithunknowns2HypothesisTesting9.1 NullandHypothesistesting-1Researchersusuallycollectdatafromasampleandthenusethesampledatatohelpanswerquestionsaboutthepopulation.Hypothesistestingisaninferentialstatisticalprocessthatuseslimitedinformationfromthesampledataastoreachageneralconclusionaboutthepopulation.3Hypothesistesting-1ResearcherAhypothesistestisaformalizedprocedurethatfollowsastandardseriesofoperations.Inthisway,researchershaveastandardizedmethodforevaluatingtheresultsoftheirresearchstudies.4Hypothesistesting-2Ahypothesistestisaformali5Thebasicexperimentalsituationforusinghypothesistestingispresentedhere.Itisassumedthattheparameterisknownforthepopulationbeforetreatment.Thepurposeoftheexperimentistodeterminewhetherornotthetreatmenthasaneffect.Isthepopulationmeanaftertreatmentthesameasordifferentfromthemeanbeforetreatment?Asampleisselectedfromthetreatedpopulationtohelpanswerthisquestion.5ThebasicexperimentalsituatProceduresofhypothesis-testing61.

First,westateahypothesisaboutapopulation.Usuallythehypothesisconcernsthevalueofapopulationparameter.Forexample,wemighthypothesizethatthemeanIQforUICstudentsism=110.2.

Next,weobtainarandomsamplefromthepopulation.Forexample,wemightselectarandomsampleofn=100UICstudents.3.

Finally,wecomparethesampledatawiththehypothesis.Ifthedataareconsistentwiththehypothesis,wewillconcludethatthehypothesisisreasonable.Butifthereisabig

discrepancybetweenthedataandthehypothesis,wewilldecidethatthehypothesisiswrong.Proceduresofhypothesis-testiNullandAlternativeHypothesesThenullhypothesis,denotedH0,isastatementofthebasicpropositionbeingtested.Itgenerallyrepresentsthestatusquo(astatementof“noeffect”or“nodifference”,orastatementofequality)andisnotrejectedunlessthereisconvincingsampleevidencethatitisfalse.The(scientificor)alternativehypothesis,denotedHa(orH1),isanalternative(tothenullhypothesis)statementthatwillbeacceptedonlyifthereisconvincingsampleevidencethatitistrue.Thesetwohypothesesaremutuallyexclusiveandexhaustive.7NullandAlternativeHypothese8Determinedbythelevelofsignificanceorthealphalevel8Determinedbythelevelofsi9Alphalevelof.05--theprobabilityofrejectingthenullhypothesiswhenitistrueisnomorethan5%.Z9Alphalevelof.05--thepro10Thelocationsofthecriticalregionboundariesforthreedifferentlevelsofsignificance10Thelocationsofthecritica11Example:Alcoholappearstobeinvolvedinavarietyofbirthdefects,includinglowbirthweightandretardedgrowth.Aresearcherwouldliketoinvestigatetheeffectofprenatalalcoholonbirthweight.Arandomsampleofn=16pregnantratsisobtained.Themotherratsaregivendailydosesofalcohol.Atbirth,onepupisselectedfromeachlittertoproduceasampleofn=16newbornrats.Theaverageweightforthesampleis15grams.Theresearcherwouldliketocomparethesamplewiththegeneralpopulationofrats.Itisknownthatregularnewbornrats(notexposedtoalcohol)haveanaverageweightofm=18grams.Thedistributionofweightsisnormalwithsd=4.11Example:12H0:μ=18

12H0:μ=18131.StatethehypothesesThenullhypothesisstatesthatexposuretoalcoholhasnoeffectonbirthweight.Thealternativehypothesisstatesthatalcoholexposuredoesaffectbirthweight.2.SelecttheLevelofSignificance(alpha)levelWewilluseanalphalevelof.05.Thatis,wearetakinga5%riskofcommittingaTypeIerror,or,theprobabilityofrejectingthenullhypothesiswhenitistrueisnomorethan5%.3.Setthedecisioncriteriabylocatingthecriticalregion131.Statethehypotheses14Alphalevelof.05--theprobabilityofrejectingthenullhypothesiswhenitistrueisnomorethan5%.Z14Alphalevelof.05--thepr154.COLLECTDATAandCOMPUTESAMPLESTATISTICSThesamplemeanisthenconvertedtoaz-score,whichisourteststatistic.5.ArriveatadecisionRejectthenullhypothesis

154.COLLECTDATAandCOMPUTEHypothesisTestingHypothesisTestingAlternativeHypothesisH1:AstatementthatisacceptedifH0isfalseWithout“=”signSay,“2”or“<2”NullHypothesisH0:

Astatementaboutthevalueofapopulationparameter(and).With“=”signSay,“=2”or“2”17Step1:Statethenullandalternate hypothesesAlternativeHypothesisH1:NulThreepossibilitiesregardingmeansH0:

m=m0H1:

m=m0H0:

m

<

m0H1:

m>m0H0:

m

>

m0H1:

m<m0Thenullhypothesisalwayscontainsequality.3hypothesesaboutmeans18aconstantStep1:Statethenullandalternate hypothesesThreepossibilitiesregardingStepTwo:SelectaLevelofSignificance,MeasuresthemaxprobabilityofrejectingatruenullhypothesisH0

isactuallytrue

butyourejectit(falsepositive).H0isfalsebutyouacceptit(falsenegative).LevelofSignificance,TypeIErrorTypeIIError19

toohighLevelofSignificance:themaximumallowableprobabilityofmakingatypeIerrorStepTwo:SelectaLevelofS

Researcher

NullAcceptsRejectsHypothesisHo

HoHoistrueHoisfalseCorrectdecisionTypeIerror(<a)TypeIIErrorCorrectDecisionRisktable20StepTwo:SelectaLevelofSignificance,

Step3:SelecttheteststatisticAteststatisticisusedtodeterminewhethertheresultoftheresearchstudy(thedifferencebetweenthesamplemeanandthepopulationmean)ismorethanwouldbeexpectedbychancealone.WewillonlyconsiderstatisticsZort,forthetimebeing.Sinceourhypothesisisaboutthepopulationmean.21Step3:SelecttheteststatisTestStatisticThetermteststatisticsimplyindicatesthatthesamplemeanisconvertedintoasingle,specificstatisticthatisusedtotestthehypotheses.Thez-scorestatisticthatisusedinthehypothesistestisthefirstspecificexampleofwhatiscalledateststatistic.Wewillintroduceseveralotherteststatisticsthatareusedinavarietyofdifferentresearchsituationslater.22TestStatisticThetermteststRejecttheH0if

Computedz

>CriticalzComputedz

<-CriticalzDecisionRuleH0:0Computedz

>CriticalzOr

Computedz<-CriticalzH0:0H0:=023DeterminedbylevelofsignificanceStep4:Formulatethedecisionrule.RejecttheH0ifComputedzCriticalvalue:

ThedividingpointbetweentheregionwhereH0isrejectedandtheregionwhereH0isaccepted,determinedbylevelofsignificance.Fromthetable,withstatisticz,onetailedtestandsignificancelevel0.05,wefoundthecriticalvalue1.65.24H0:0Rejectifz

>CriticalzCriticalvalue:ThedividingpOne-TailedTestofSignificance.IfH0:0istrue,itisveryunlikelythatthecomputedzvalueissolarge.25One-TailedTestofSignificanc26H0:0Computedz

<-CriticalzRejecttheH0ifIfH0:0istrue,itisveryunlikelythatthecomputedzvalue(fromthesamplemean)issosmall.26H0:0Computedz<-IfH0:=0istrue,itisveryunlikelythatthecomputedzvalueisextremelylargeorsmall.Two-TailedTestsofSignificance27IfH0:=0istrue,itisvStep5:Makeadecision.28Reject!Accept!Step5:Makeadecision.28RejeAninsurancecompanyisreviewingitscurrentpolicyrates.Whenoriginallysettingtheratestheybelievedthattheaverageclaimamountwas$1,800.Theyareconcernedthatthetruemeanisactuallyhigherthanthis,becausetheycouldpotentiallylosealotofmoney.Theyrandomlyselect40claims,andcalculateasamplemeanof$1,950.Assumingthatthepopulationstandarddeviationofclaimsis$500,andsetlevelofsignificance

=0.05,testtoseeiftheinsurancecompanyshouldbeconcerned.29Step1:SetthenullandalternativehypothesesExampleOneTailed(UpperTailed)Aninsurancecompanyisreview30Step2:CalculatetheteststatisticExampleOneTailed(UpperTailed)Step3:SetRejectionRegionLookingatthepicturebelow,weneedtoputallofalphaintherighttail.Thus,R:Z>1.9630Step2:Calculatethetests31Step4:ConcludeWecanseethatz=1.897<1.96,thusourteststatisticisnotintherejectionregion.Thereforewefailtorejectthenullhypothesis.

Wecannotconcludeanythingstatisticallysignificantfromthistest,andcannottelltheinsurancecompanywhetherornottheyshouldbeconcernedabouttheircurrentpolicies.ExampleOneTailed(UpperTailed)31Step4:ConcludeExampleOne32Tryingtoencouragepeopletostopdrivingtocampus,theuniversityclaimsthatonaverageittakespeople30minutestofindaparkingspaceoncampus.Johndoesnotthinkittakessolongtofindaspot.Hecalculatedthemeantimetofindaparkingspaceoncampusforthelastfivetimesandfoundittobe20minutes.Assumingthatthetimeittakestofindaparkingspotisnormallydistributed,andthatthepopulationstandarddeviation=6minutes,performahypothesistestwithlevelofsignificancealpha=0.10toseeifhisclaimiscorrect.Example:OneTailed(LowerTailed)32Tryingtoencouragepeoplet33Step1:SetthenullandalternativehypothesesExample:OneTailed(LowerTailed)Step2:CalculatetheteststatisticStep3:SetRejectionRegionLookingatthepicturebelow,weneedtoputallofalphainthelefttail.Thus,R:Z<-1.2833Step1:Setthenullandalt34Example:OneTailed(LowerTailed)Step4:ConcludeWecanseethatz=-3.727<-1.28,thusourteststatisticisintherejectionregion.Thereforewerejectthenullhypothesisinfavorofthealternative.Weconcludethatthemeanissignificantlylessthan30,thusJohnhasproventhatthemeantimetofindaparkingspaceislessthan30.34Example:OneTailed(LowerT35Example:TwoTailedAsampleof40salesreceiptsfromagrocerystorehasmean

=$137and

populationstandarddeviation

=$30.2.Usethesevaluestotestwhetherornotthemeaninsalesatthegrocerystorearedifferentfrom$150withlevelofsignificancealpha=0.01.Step1:SetthenullandalternativehypothesesStep2:Calculatetheteststatistic35Example:TwoTailedAsample36Example:TwoTailedStep3:SetRejectionRegionLookingatthepicturebelow,weneedtoputhalfofalphainthelefttail,andtheotherhalfofalphaintherighttail.Thus, R:Z<-2.58orZ>2.58Step4:ConcludeWeseethatZ=-2.722<-2.58,thusourteststatisticisintherejectionregion.Thereforewerejectthenullhypothesisinfavorofthealternative.Wecanconcludethatthemeanissignificantlydifferentfrom$150,thusIhaveproventhatthemeansalesatthegrocerystoreisnot$150.36Example:TwoTailedStep3:SExample:creditmanagerLisa,thecreditmanager,wantstocheckifthemeanmonthlyunpaidbalanceismorethan$400.Thelevelofsignificanceshesetis.05.Arandomcheckof172unpaidbalancesrevealedthesamplemeantobe$407.Thepopulationstandarddeviationisknowntobe$38.ShouldLisaconcludethatthepopulationmeanisgreaterthan$400,orisitreasonabletoassumethatthedifferenceof$7($407-$400)isduetochance?(atconfidencelevel0.05)37Example:creditmanagerLisa,tStep1H0:μ<$400H1:μ>$400Step2Thesignificancelevelis.05.Step3Sinceisknown,wecanfindtheteststatisticz.Step4H0isrejectedifz>1.65(since=0.05)Step5Makeadecisionandinterprettheresults.(Nextpage)Example:Lisa,thecreditmanager38Step1Step2Step3Step4StepThep-valueis.0078foraone-tailedtest.(reftoinformalans.)Computedzof2.42>Criticalz

of1.65,pof.0078<aof.05.

RejectH0.Step5Makeadecisionandinterprettheresults.Wecanconcludethatthemeanunpaidbalanceisgreaterthan$400.39Thep-valueis.0078foraoneLimitationofz-scoresinhypothesistestingThelimitationofz-scoresinhypothesistestingisthatthepopulationstandarddeviation(orvariance)mustbeknown.Whatifyoudon’tknowtheμand

ofthepopulation?Answer:usethesamplevariabilityinstead40Limitationofz-scoresinhypo41Samplevariances2=sumofsquaresofdeviation/(n-1) =sumofsquareofdeviations/df =SS/dfSinceyoumustknowthesamplemeanbeforeyoucancomputesamplevariance,thisplacesarestrictiononsamplevariabilitysuchthatonlyn-1scoresinasamplearefreetovary.Thevaluen-1iscalledthedegreesoffreedom(ordf)forthesamplevariance.41SamplevarianceSinceyoumu42Ifyouselectallthepossiblesamplesofaparticularsize(n),thesetofallpossibletstatisticswillformatdistribution.ZstatistictstatisticUnknown

Goodfor: (i)largesamplen>30,withtheunderlyingdistributionmayormaynotbeNormal (ii)smallsamplen<30withtheunderlyingdistributionisNormal42Ifyouselectallthepossib43Distributionsofthetstatisticfordifferentvaluesofdegreesoffreedomarecomparedtoanormaldistribution.43Distributionsofthetstati44444545464647Thetdistributionwithdf=3.Notethat5%ofthedistributionislocatedinthetailst>2.353andt<2.353.47Thetdistributionwithdf=ThelabelonFries’Catsupindicatesthatthebottlecontains16ouncesofcatsup.Asampleof36bottlesfromlasthour’sproductionrevealedameanweightof16.12ouncesperbottleandasamplestandarddeviationof0.5ounces.Atthe0.05significancelevel,testiftheprocessoutofcontrol?Thatis,canweconcludethatthemeanamountperbottleisdifferentfrom16ounces?48ThelabelonFries’CatsupindStep1Statethenullandthealternativehypotheses

H0:m=16 H1:m=16Step3Sincethesamplesizeislargeenoughandthepopulations.d.isunknown,wecanusetheteststatisticist.Step2Selectthesignificancelevel.Thesignificancelevelis.05.Step4Statethedecisionrule.RejectH0ifz>1.96

orz<-1.96(since=0.05)Step5Makeadecisionandinterprettheresults.(Nextpage)49Step1Step3Step2Step4SteComputedzof1.44<Criticalz

of1.96,pof.1499>aof.05,Donotrejectthenullhypothesis.Thep-valueis.1499foratwo-tailedtest.Step5:

Makeadecisionandinterprettheresults.Wecannotconcludethemeanisdifferentfrom16ounces.50Computedzof1.44Thep-valueTheteststatisticisthetdistribution.TestingforaPopulationMean:Unknown(Population)standarddeviation,Smallsample.ButtheunderlyingdistributionisNormalThecriticalvalueoftisdeterminedbyitsdegreesoffreedomwhichisequalton-1.51TheteststatisticisthetdiThecurrentrateforproducing5ampfusesataElectricCo.is250perhour.Anewmachinehasbeenpurchasedandinstalled.Accordingtothesupplier,theproductionratearenormallydistributed.Asampleof10randomlyselectedhoursfromlastmonthrevealedthatthemeanhourlyproductionwas256units,withasamples.d.of6perhour.

Atthe0.05significancelevel,testifthenewmachineisfasterthantheoldone?52ThecurrentrateforproducingStep1Statethenullandalternatehypotheses.H0:μ<

250H1:μ>250Step2

Selectthelevelofsignificance.Itis.05.Step3Sincetheunderlyingdistributionisnormal,sisunknown,usethetdistribution.Step4Statethedecisionrule.degreesoffreedom=10–1=9.RejectH0ift>1.83353Step1Step2Step3Step4Computedtof3.162>Criticalt

of1.833pof.0058<alphaof.05RejectHoThep-valueis0.0058.(obtainedfromt,needasoftwaretofindit.)Step5Makeadecisionandinterprettheresults.Themeannumberoffusesproducedismorethan250perhour.54Ifthep-valueislessthanalpha,thenrejectthenullhypothesis.Computedtof3.162Thep-valuAmountoftimeUICstudentsspendinlibraryfromsurveyMean41.72minutesStandarddeviation40.179minutesNumberofcases294Nationalsurveyfindsuniversitylibraryusersspendmeanof38minutesIspopulationmeanforUICLibraryusersdifferentfromnationalmean?Example:One-samplehypothesistestformeanAmountoftimeUICstudentsspNullhypothesis

H0:μ=μ0

μ=38

Alternativeorresearchhypothesis

Ha:μ≠μ0

μ≠38Step1.HypothesesNullhypothesis

H0:μ=μ0 Step2.LevelofsignificanceProbabilityoferrorinmakingdecisiontorejectnullhypothesisForthistestchoose

α=0.05Step2.LevelofsignificancePStep3.Teststatisticn=294sousecriticaltvaluesfromtableforinfinity.Step3.Teststatisticn=294CannotrejectthenullhypothesisCannotconcludethatpopulationmeanisdifferentfrom38minutes4.Decision95%confidenceIntervalinthisexample:E=1.96* =4.59[41.72-4.59,41.72+4.59]or[37.13,46.31]CannotrejectthenullhypotheConfidenceintervalfortimespentinlibraryis37.13<μ<46.31Hypothesizedvalueof38minutesfallswithinconfidenceintervalThereforewecannotsaythatpopulationmeanisnotequalto38minutes,cannotrejectthenullhypothesisConfidenceintervalandhypothesistestforlibraryexampleConfidenceintervalfortimesForparametersforasinglesample…One-samplehypothesistestinvolvescomparisonwithpre-specifiedvalue…Whichisoftenartificial…SoconfidenceintervalmostappropriateforreportingresultsForparametersfortwosamples…DifferenceinparametersisofinterestHypothesistestexaminesdirectlyConfidenceintervallessintuitiveUsingconfidenceintervalsorhypothesistestsForparametersforasinglesaConfidenceinterval

orHypothesis

test?Hypothesistestsarebetterwhenthechiefissueistomakeayes/nodecisionaboutwhetherapatternexistsinapopulation.Confidenceintervalsarebetterwhenthechiefissueistomakeabestguessofapopulationparameter.Confidenceinterval

orHypoth63Whenreadingascientificjournal,youtypicallywillnotbetoldexplicitlythattheresearcherevaluatedthedatausingaz-scoreasateststatisticwithanalphalevelof.05.Norwillyoubetoldthat“thenullhypothesisisrejected.”Instead,youwillseeastatementsuchas:Thetreatmentwithmedicationhadasignificanteffectonpeople’sdepressionscores,z=3.85,p<.05.Letusexaminethisstatementpiecebypiece.First,whatismeantbythetermsignificant?Instatisticaltests,thiswordindicatesthattheresultisdifferentfromwhatwouldbeexpectedduetochance.Asignificantresultmeansthatthenullhypothesishasbeenrejected.Thatis,thedataaresignificantbecausethesamplemeanfallsinthecriticalregionandisnotwhatwewouldhaveexpectedtoobtainifH0weretrue.Next,whatisthemeaningofz=3.85?Thezindicatesthataz-scorewasusedastheteststatistictoevaluatethesampledataandthatitsvalueis3.85.63Whenreadingascientificjo64Finally,whatismeantbyp<.05?Thispartofthestatementisaconventionalwayofspecifyingthealphalevelthatwasusedforthehypothesistest.Morespecifically,wearebeingtoldthatanoutcomeasextremeastheresultoftheexperimentwouldoccurbychancewithaprobability(p)thatislessthan.05(alpha)ifH0weretrue.64Finally,whatismeantbyp<65IncircumstanceswherethestatisticaldecisionistofailtorejectH0,thereportmightstatethatTherewasnoevidencethatthemedicationhadaneffectondepressionscores,z=1.30,p>.05.Inthiscase,wearesayingthattheobtainedresult,z=1.30,isnotunusual(notinthecriticalregion)andisrelativelylikelytooccurbychance(theprobabilityisgreaterthan.05).Thus,H0wasnotrejected.65IncircumstanceswherethesUsingthep-ValueinHypothesisTestingIfthep-Valuea,H0cannotberejected.Ifthep-Value<a,H0isrejected.p-valuedoesnotonlytelluswhetherweshouldrejectH0,butalsotellushowconfidentwearetorejectit.66Samplemeansthatfallinthecriticalregion(shadedareas)haveaprobabilitylessthanalpha.H0shouldberejected.Usingthep-ValueinHypothesi67MoreExample:Totesttheeffectivenessofeye-spotpatternsindeterringpredation,asampleofn=16insectivorousbirdsisselected.Theanimalsaretestedinaboxthathastwoseparatechambers(seefigure).Thebirdsarefreetoroamfromonechambertoanotherthroughadoorwayinapartition.Onthewallofonechamber,twolargeeye-spotpatternshavebeenpainted.Theotherchamberhasplainwalls.Thebirdsaretestedoneatatimebyplacingtheminthedoorwayinthecenteroftheapparatus.Eachanimalisleftintheboxfor60minutes,andtheamountoftimespentintheplainchamberisrecorded.Supposethatthesampleofn=16birdsspentanaveragemof39minutesintheplainside,withSS=540.Canweconcludethateye-spotpatternshaveaneffectonbehavior?Notethatwehavenoinformationaboutthepopulationvariance.67MoreExample:Totesttheef68Step1:Statethehypotheses:H0:μplainside=30minutesStep2:Locatethecriticalregion.Theteststatisticisatstatisticbecausethepopulationvarianceisnotknown. df=16-1=15Foratwo-tailedtestatthe.05levelofsignificanceandwith8degreesoffreedom,thecriticalregionconsistsoftvaluesgreaterthan+2.131orlessthan-2.131Step3:Calculatetheteststatistic s2=SS/df=540/15=36 sm=sqrt(s2/16)=1.5

thetstatistict=(39-30)/1.5=6Step4:Makeadecision–rejectH068Step1:Statethehypotheses69Thecriticalregioninthetdistributionforalpha=.05anddf=15.69Thecriticalregioninthet70HYPOTHESISTESTINGfor:populationproportions70HYPOTHESISTESTINGfor:Example:Surveydataonattitudestoward

incomeinequalityImaginethatwewouldliketofindoutifUSadultshadsomenetopiniononthefollowingissue.“Doyouthinkitshouldorshouldnotbethegovernment’sresponsibilitytoreduceincomedifferencesbetweentherichandthepoor?”Score Response Number1 shouldbe 5910 shouldnotbe 636Totaln=1227Example:SurveydataonattituSurveydataonattitudestoward

incomeinequality0:Assumptions:wewillbedoingalarge-sampletestforpopulationproportions.Toperformthistest,wemustassumethat…Samplesizeislargeenoughthatnp(1-p)>10

ThesampleisarandomsampleofsomesortThevariableisadiscreteinterval-scalevariable,whichisautomaticallytrueforpopulationproportions.SurveydataonattitudestowarSurveydataonattitudestoward

incomeinequality1:Hypothesis:

letdenotethepopulationproportionwhofavorgovernmentinterventiontoalleviateincomeinequality.Ournullhypothesisisthatthepopulation,onaverage,neithersupportsnoropposesgovernmentintervention.Ho:=0.5ThealternatehypothesisisthenHA:0.5SurveydataonattitudestowarSurveydataonattitudestoward

incomeinequality2:TestStatistic:Forannof1227respondents,wecalculatethefollowingstatistics:P =n(yes)/n(total)=591/1227=.4817σ0 =SQRT(o(1-o))=.5SE =σ0/SQRT(n)=.01427z =(P-o

)/s.e.. =(.4817-.500)/.01427. =-1.282Thez-statisticistheteststatisticofinterestinalarge-sampletestofapopulationproportion.SurveydataonattitudestowarSurveydataonattitudestoward

incomeinequality3.Pickα=0.05&determinecriticalz-1.282SurveydataonattitudestowarHypothesisTesting統(tǒng)計(jì)學(xué)假設(shè)檢驗(yàn)76HypothesisTesting統(tǒng)計(jì)學(xué)假設(shè)檢驗(yàn)1HypothesisTesting9.1 NullandAlternativeHypothesesandErrorsinTesting9.2 zTestsaboutaPopulationwithknowns9.3 tTestsaboutaPopulationwithunknowns77HypothesisTesting9.1 NullandHypothesistesting-1Researchersusuallycollectdatafromasampleandthenusethesampledatatohelpanswerquestionsaboutthepopulation.Hypothesistestingisaninferentialstatisticalprocessthatuseslimitedinformationfromthesampledataastoreachageneralconclusionaboutthepopulation.78Hypothesistesting-1ResearcherAhypothesistestisaformalizedprocedurethatfollowsastandardseriesofoperations.Inthisway,researchershaveastandardizedmethodforevaluatingtheresultsoftheirresearchstudies.79Hypothesistesting-2Ahypothesistestisaformali80Thebasicexperimentalsituationforusinghypothesistestingispresentedhere.Itisassumedthattheparameterisknownforthepopulationbeforetreatment.Thepurposeoftheexperimentistodeterminewhetherornotthetreatmenthasaneffect.Isthepopulationmeanaftertreatmentthesameasordifferentfromthemeanbeforetreatment?Asampleisselectedfromthetreatedpopulationtohelpanswerthisquestion.5ThebasicexperimentalsituatProceduresofhypothesis-testing811.

First,westateahypothesisaboutapopulation.Usuallythehypothesisconcernsthevalueofapopulationparameter.Forexample,wemighthypothesizethatthemeanIQforUICstudentsism=110.2.

Next,weobtainarandomsamplefromthepopulation.Forexample,wemightselectarandomsampleofn=100UICstudents.3.

Finally,wecomparethesampledatawiththehypothesis.Ifthedataareconsistentwiththehypothesis,wewillconcludethatthehypothesisisreasonable.Butifthereisabig

discrepancybetweenthedataandthehypothesis,wewilldecidethatthehypothesisiswrong.Proceduresofhypothesis-testiNullandAlternativeHypothesesThenullhypothesis,denotedH0,isastate