NominalData

NominalDataGregCElversParametricStatisticsTheinferentialstatisticsthatwehavediscussed,suchastandANOVA,areparametricstatisticsAparametricstatisticisastatisticthatmakescertainassumptionsabouthowthedataaredistributedTypically,theyassumethatthedataaredistributednormallyNonparametricStatisticsNonparametricstatisticsdonotmakeassumptionsabouttheunderlyingdistributionofthedataThus,nonparametricstatisticsareusefulwhenthedataarenotnormallydistributedBecausenominallyscaledvariablescannotbenormallydistributed,nonparametricstatisticsshouldbeusedwiththemParametricvsNonparametricTestsWhenyouhaveachoice,youshoulduseparametricstatisticsbecausetheyhavegreaterstatisticalpowerthanthecorrespondingnonparametrictestsThatis,parametricstatisticsaremorelikelytocorrectlyrejectH0thannonparametricstatisticsBinomialTestThebinomialtestisatypeofnonparametricstatisticThebinomialtestisusedwhentheDVisnominal,andithasonlytwocategoriesorclassesItisusedtoanswerthequestion:Inasample,istheproportionofobservationsinonecategorydifferentthanagivenproportion?BinomialTestAresearcherwantstoknowiftheproportionofailurophilesinagroupof20librariansisgreaterthanthatfoundinthegeneralpopulation,.40Thereare9ailurophilesinthegroupof20librariansBinomialTestWriteH0andH1:H0:P£.40H1:P>.40Isthehypothesisone-tailedortwo-tailed?Directional,one-tailedDeterminethestatisticaltestThelibrarianscaneitherbeornotbeailurophiles,thuswehaveadichotomous,nominallyscaledvariableUsethebinomialtestBinomialTestDeterminethecriticalvaluefromatableofcriticalbinomialvaluesFindthecolumnthatcorrespondstothepvalue(inthiscase.40)Findtherowthatcorrespondstothesamplesize(N=20)anda(.05)Thecriticalvalueis13BinomialTestIftheobservednumberofailurophiles(9)isgreaterthanorequaltothecriticalvalue(13),youcanrejectH0WefailtorejectH0;thereisinsufficientevidencetoconcludethatthepercentageoflibrarianswhoareailurophilesisprobablygreaterthanthatofthegeneralpopulationNormalApproximationtotheBinomialTestWhenthesamplesizeisgreaterthanorequalto50,thenanormalapproximation(i.e.az-test)canbeusedinplaceofthebinomialtestWhentheproductofthesamplesize(N),p,and1-pisgreaterthanorequalto9,thenthenormalapproximationcanbeuseNormalApproximationtotheBinomialTestThenormalapproximationtothebinomialtestisdefinedas:x=numberofobservationsinthecategoryN=samplesizeP=probabilityinquestionNormalApproximationtotheBinomialTestAresearcherwantstoknowiftheproportionofailurophilesinagroupof100librariansisgreaterthanthatfoundinthegeneralpopulation,.40Thereare43ailurophilesinthegroupof100librariansNormalApproximationtotheBinomialTestWriteH0andH1:H0:P£.40H1:P>.40Isthehypothesisone-tailedortwo-tailed?Directional,one-tailedDeterminethestatisticaltestThelibrarianscaneitherbeornotbeailurophiles,thuswehaveadichotomous,nominallyscaledvariableUsetheztest,becausen³50NormalApproximationtotheBinomialTestCalculatethez-scoreNormalApproximationtotheBinomialTestDeterminethecriticalvaluefromatableofareaunderthenormalcurveFindthez-scorethatcorrespondstoanareaof.05abovethez-scoreThatvalueis1.65Comparethecalculatedz-scoretothecriticalz-scoreIf|zcalculated|³zcritical,thenrejectH00.612<1.65;failtorejectH0c2--OneVariableWhenyouhavenominaldatathathasmorethantwocategories,thebinomialtestisnotappropriateThec2(chisquared)testisappropriateinsuchinstancesThec2testanswersthefollowingquestion:Istheobservednumberofitemsineachcategorydifferentfromatheoreticallyexpectednumberofobservationsinthecategories?c2--OneVariableAtarecentGREtest,eachof28studentstookoneof5subjecttestsWasthereanequalnumberoftesttakersforeachtest?c2--OneVariableWriteH0andH1:H0:S(O-E)2=0H1:S(O-E)2¹0O=observedfrequenciesE=expectedfrequenciesSpecifyaa=.05Calculatethec2statisticc2=S[(Oi-Ei)2/Ei]c2Calculationsc2Calculationsc2=S[(Oi-Ei)2/Ei]c2=7.31+2.31+0.46+0.29+0.46=10.83Calculatethedegreesoffreedom:df=numberofgroups-1=5-1=4Determinethecriticalvaluefromatableofcriticalc2valuesdf=4,a=.05Criticalc2a=.05(4)=9.488c2DecisionIftheobserved/calculatedvalueofc2isgreaterthanorequaltothecriticalvalueofc2,thenyoucanrejectH0thatthereisnodifferencebetweentheobservedandexpectedfrequenciesBecausetheobservedc2=10.83islargerthanthecriticalc2=9.488,wecanrejectH0thattheobservedandexpectedfrequenciesarethesamec2TestofIndependencec2canalsobeusedtodetermineiftwovariablesareindependentofeachotherE.g.,isbeinganailurophileindependentofwhetheryouaremaleorfemale?WriteH0andH1:H0:SS(O-E)2=0H1:SS(O-E)2¹0Specifyaa=.05c2TestofIndependenceTheprocedureforansweringsuchquestionsisvirtuallyidenticaltotheonevariablec2procedure,exceptthatwehavenotheoreticalbasisfortheexpectedfrequenciesTheexpectedfrequenciesarederivedfromthedatac2TestofIndependenceTheexpectedfrequenciesaregivenbytheformulatotheright:c2TestofIndependencec2TestofIndependenceCalculatetheobservedvalueofc2c2TestofIndependenceFirst,determinethedegreesoffreedom:df=(r-1)*(c-1)Inthisexample,thenumberofrows(r)is2,andthenumberofcolumns(c)is2,sothedegreesoffreedomare(2-1)*(2-1)=1Determinethecriticalvalueofc2fromatableofcriticalc2valuesCriticalc2a=.05(1)=3.841c2TestofIndependenceMakethedecisionIftheobserved/calculatedvalueofc2isgreaterthanorequaltothecriticalvalueofc2,thenyoucanrejectH0thattheexpectedandobservedfrequenciesareequalIfthisexample,theobservedc2=3.319isnotgreaterthanorequaltothecriticalc2=3.841,sowefailtorejectH0RequirementsfortheUseofc2Eventhoughc2makesnoassumptionsabouttheunderlyingdistribution,itdoesmakesomeassumptionsthatneedstobemetpriortouseAssumptionofindependenceFrequenciesmustbeused,notpercentagesSufficientlylargesamplesizeAssumptionofIndependenceEachobservationmustbeunique;thatisanindividualcannotbecontainedinmorethanonecategory,orcountedinonecategorymorethanonceWhenthisassumptionisviolated,theprobabilityofmakingaType-IerrorisgreatlyenhancedFrequenciesThedatamustcorrespondtofrequenciesinthecategories;percentagesarenotappropriateasdataSufficientSampleSizeDifferentpeoplehavedifferentrecommendationabouthowlargethesampleshouldbe,andwhattheminimumexpectedfrequencyineachcellshouldbeGood,Grover,andMitchell(1977)suggestthattheexpectedfrequenciescanbeaslowas0.33withoutincreasingthelikelihoodofmakingaType-IerrorSmallsamplesreducepower