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Usingtheoreticalsamplingprobabilitydistributions
Samplingdistributionsallowustoapproximatetheprobabilitythataparticularvaluewouldoccurbychancealone.Ifyoucollectedmeansfromaninfinitenumberofrepeatedrandomsamplesofthesamesamplesizefromthesamepopulationyouwouldfindthatmostmeanswillbeverysimilarinvalue,inotherwords,theywillgrouparoundthetruepopulationmean.Mostmeanswillcollectaboutacentralvalueormidpointofasamplingdistribution.Thefrequencyofmeanswilldecreaseasonetravelsawayfromthecenterofanormalsamplingdistribution.Inanormalprobabilitydistribution,about95%ofthemeansresultingfromaninfinitenumberofrepeatedrandomsampleswillfallbetween1.96standarderrorsaboveandbelowthemidpointofthedistributionwhichrepresentsthetruepopulationmeanandonly5%willfallbeyond(2.5%ineachtailofthedistribution).
Thefollowingarecommonlyusedpointsonadistributionfordecidingstatisticalsignificance:
90%ofscores+/-1.65standarderrors
95%ofscores+/-1.96standarderrors
99%ofscores+/-2.58standarderrors
Standarderror:
Mathematicaladjustmenttothestandarddeviationtoaccountfortheeffectsamplesizehasontheunderlyingprobabilitydistribution.Itrepresentsthestandarddeviationofthesamplingdistribution
Alphaandtheroleofthedistributiontails
Thepercentageofscoresbeyondaparticularpointalongthexaxisofasamplingdistributionrepresentthepercentofthetimeduringaninfinitenumberofrepeatedsamplesonewouldexpecttohaveascoreatorbeyondthatvalueonthexaxis.Thisvalueonthexaxisisknownasthecriticalvaluewhenusedinhypothesistesting.Themidpointrepresentstheactualpopulationvalue.Mostscoreswillfallneartheactualpopulationvaluebutwillexhibitsomevariationduetosamplingerror.Ifascorefromarandomsamplefalls1.96standarderrorsorfartheraboveorbelowthemeanofthesamplingdistribution,weknowfromtheprobabilitydistributionthatthereisonlya5%chanceofrandomlyselectingasetofscoresthatwouldproduceasamplemeanthatfarfromthetruepopulationmean.Whenconductingsignificancetesting,ifwehaveateststatisticthatis1.96standarderrorsaboveorbelowthemeanofthesamplingdistribution,weassumewehaveastatisticallysignificantdifferencebetweenoursamplemeanandtheexpectedmeanforthepopulation.Sinceweknowavaluethatfarfromthepopulationmeanwillonlyoccurrandomly5%ofthetime,weassumethedifferenceistheresultofatruedifferencebetweenthesampleandthepopulationmean,andisnottheresultofrandomsamplingerror.The5%isalsoknownasalphaandistheprobabilityofbeingwrongwhenweconcludestatisticalsignificance.
1-tailedvs.2-tailedstatisticaltests
A2-tailedtestisusedwhenyoucannotdetermineaprioriwhetheradifferencebetweenpopulationparameterswillbepositiveornegative.A1-tailedtestisusedwhenyoucanreasonablyexpectadifferencewillbepositiveornegative.Ifyouretainthesamecriticalvaluefora1-tailedtestthatwouldbeusedifa2-tailedtestwasemployed,thealphaishalved(i.e.,.05alphawouldbecome.025alpha).
HypothesisTesting
Thechainofreasoningandsystematicstepsusedinhypothesistestingthatareoutlinedinthissectionarethebackboneofeverystatisticaltestregardlessofwhetheronewritesouteachstepinaclassroomsettingorusesstatisticalsoftwaretoconductstatisticaltestsonvariablesstoredinadatabase.
Chainofreasoningforinferentialstatistics
1.Sample(s)mustberandomlyselected
2.Sampleestimateiscomparedtounderlyingdistributionofthesamesizesamplingdistribution
3.Determinetheprobabilitythatasampleestimatereflectsthepopulationparameter
Thefourpossibleoutcomesinhypothesistesting
ActualPopulationComparison
NullHyp.True
NullHyp.False
DECISION
(thereisnodifference)
(thereisadifference)
RejectedNullHyp
TypeIerror
(alpha)
CorrectDecision
DidnotRejectNull
TypeIIError
(Alpha=probabilityofmakingaTypeIerror)
Regardlessofwhetherstatisticaltestsareconductedbyhandorthroughstatisticalsoftware,thereisanimplicitunderstandingthatsystematicstepsarebeingfollowedtodeterminestatisticalsignificance.Thesegeneralstepsaredescribedonthefollowingpageandinclude1)assumptions,2)statedhypothesis,3)rejectioncriteria,4)computationofstatistics,and5)decisionregardingthenullhypothesis.Theunderlyinglogicisbasedonrejectingastatementofnodifferenceornoassociation,calledthenullhypothesis.Thenullhypothesisisonlyrejectedwhenwehaveevidencebeyondareasonabledoubtthatatruedifferenceorassociationexistsinthepopulation(s)fromwhichwedrewourrandomsample(s).
Reasonabledoubtisbasedonprobabilitysamplingdistributionsandcanvaryattheresearcher'
sdiscretion.Alpha.05isacommonbenchmarkforreasonabledoubt.Atalpha.05weknowfromthesamplingdistributionthatateststatisticwillonlyoccurbyrandomchancefivetimesoutof100(5%probability).Sinceateststatisticthatresultsinanalphaof.05couldonlyoccurbyrandomchance5%ofthetime,weassumethattheteststatisticresultedbecausetherearetruedifferencesbetweenthepopulationparameters,notbecausewedrewanextremelybiasedrandomsample.
Whenlearningstatisticswegenerallyconductstatisticaltestsbyhand.Inthesesituations,weestablishbeforethetestisconductedwhatteststatisticisneeded(calledthecriticalvalue)toclaimstatisticalsignificance.So,ifweknowforagivensamplingdistributionthatateststatisticofplusorminus1.96wouldonlyoccur5%ofthetimerandomly,anyteststatisticthatis1.96orgreaterinabsolutevaluewouldbestatisticallysignificant.Inananalysiswhereateststatisticwasexactly1.96,youwouldhavea5%chanceofbeingwrongifyouclaimedstatisticalsignificance.Iftheteststatisticwas3.00,statisticalsignificancecouldalsobeclaimedbuttheprobabilityofbeingwrongwouldbemuchless(about.002ifusinga2-tailedtestortwo-tenthsofonepercent;
0.2%).Both.05and.002areknownasalpha;
theprobabilityofaTypeIerror.
Whenconductingstatisticaltestswithcomputersoftware,theexactprobabilityofaTypeIerroriscalculated.Itispresentedinseveralformatsbutismostcommonlyreportedas"
p<
"
or"
Sig."
Signif."
Significance."
Using"
asanexample,ifaprioriyouestablishedathresholdforstatisticalsignificanceatalpha.05,anyteststatisticwithsignificanceatorlessthan.05wouldbeconsideredstatisticallysignificantandyouwouldberequiredtorejectthenullhypothesisofnodifference.Thefollowingtablelinkspvalueswithabenchmarkalphaof.05:
P<
Alpha
ProbabilityofTypeIError
FinalDecision
.05
5%chancedifferenceisnotsignificant
Statisticallysignificant
.10
10%chancedifferenceisnotsignificant
Notstatisticallysignificant
.01
1%chancedifferenceisnotsignificant
.96
96%chancedifferenceisnotsignificant
StepstoHypothesisTesting
Hypothesistestingisusedtoestablishwhetherthedifferencesexhibitedbyrandomsamplescanbeinferredtothepopulationsfromwhichthesamplesoriginated.
GeneralAssumptions
∙Populationisnormallydistributed
∙Randomsampling
∙Mutuallyexclusivecomparisonsamples
∙Datacharacteristicsmatchstatisticaltechnique
Forinterval/ratiodatauseÊ
t-tests,Pearsoncorrelation,ANOVA,regression
Fornominal/ordinaldatauseÊ
Differenceofproportions,chisquareandrelatedmeasuresofassociation
StatetheHypothesis
NullHypothesis(Ho):
Thereisnodifferencebetween___and___.
AlternativeHypothesis(Ha):
Thereisadifferencebetween__and__.
Note:
Thealternativehypothesiswillindicatewhethera1-tailedora2-tailedtestisutilizedtorejectthenullhypothesis.
Hafor1-tailtested:
The__of__isgreater(orless)thanthe__of__.
SettheRejectionCriteria
Thisdetermineshowdifferenttheparametersand/orstatisticsmustbebeforethenullhypothesiscanberejected.This"
regionofrejection"
isbasedonalpha(
)--theerrorassociatedwiththeconfidencele