CHAPTER2Two-SidedTests:Introduction2.1Two-SidedTestsforComparingTwoTreatmentswithNormalResponseofKnownVarianceThenametwo-sidedtestisgiventoatestofahypothesisagainstatwo-sidedalternative.Inthischapterwerestrictattentiontotestingforadifferenceinthemeanresponseoftwotreatmentswhenobservationsarenormallydistributedwithcommon,knownvariance.Denotingthedifferenceinmeansbyθ,thenullhypothesisH0:θ=0statesthatresponsesfollowthesamedistributionunderbothtreatments.Thealternativehypothesis,HA:θ̸=0,containstwocases,θ<0andθ>0,whichcorrespondtoonetreatmentbeingsuperiortotheotherandviceversa.Inthiscomparison,thestandardizedteststatistic,Z,isdistributedsymmetricallyabout0underH0,andafixedsampletestrejectsH0if|Z|>cforsomeconstantc.ThesignofZdetermineswhichtreatmentistobepreferredwhenH0isrejected.Atest’sTypeIerrorprobabilityisdefinedtobetheprobabilityofwronglyrejectingthenullhypothesis,Prθ=0{|Z|>c}.Thepowerofatestistheprobabilityofrejectingthenullhypothesiswhenitdoesnothold,Prθ{|Z|>c},forvaluesofθ̸=0.Thepowerdependsonθ,increasingasθmovesawayfrom0,butitisconvenienttostateapowerrequirementataspecificvalueorvaluesofθ,forexample,Prθ=δ{|Z|>c}=Prθ=−δ{|Z|>c}=1−β,(2.1)whereδrepresentsatreatmentdifferencethattheinvestigatorswouldhopetodetectwithhighprobability.ThesmallprobabilityβisreferredtoastheTypeIIerrorprobabilityatθ=±δ.Inpracticalterms,itisnotdesirabletorejectH0infavorofθ<0whenactuallyθ>0asthiswouldimplyarecommendationoftheinferiorofthetwotreatments.Withthisinmind,theabovepowerrequirementcanbereplacedbyPrθ=δ{Z>c}=Prθ=−δ{Z<−c}=1−β.(2.2)Fortunately,forvaluesofcandδthatweshallconsider,theprobabilitiesPrθ=δ{Z<−c}andPrθ=−δ{Z>c}areextremelysmallandthetworequirements(2.1)and(2.2)areequalforallpracticalpurposes.Wepresentthestandardnon-sequentialtestforthisprobleminSection2.2,andintheremainderofthechapterwedescribeandcomparegroupsequentialteststhatachievespecifiedTypeIandTypeIIerrors.Thereisnolossofgeneralityc⃝2000byChapman&Hall/CRCinintroducingtwo-side...
