Homework9Dueon2018/5/181.(TextbookSection8.1-1,Page299)LetXhaveanexponentialdistributionwithpa-rameterβ.SupposethatwewishtotestthehypothesesH0:β≥1,H1:β<1.ConsiderthetestprocedureδthatrejectsH0ifX≥1.Computethesizeofthetest.2.(TextbookSection8.1-2,Page299)SupposethatX1,...,Xnformarandomsamplefromauniformdistributionontheinterval[0,θ],andthatthefollowinghypothesesaretobetested:H0:θ≥2,H1:θ<2.LetYn=max(X1,...,Xn),andconsideratestproceduresuchthatthecriticalregioncontainsalltheoutcomesforwithYn≤1.5.Determinethesizeofthetest.3.(TextbookSection8.1-3,Page299)Supposethattheproportionpofdefectiveitemsinalargepopulationofitemsisunknown,andthatitisdesiredtotestthefollowinghypotheses:H0:p=0.2,H1:p̸=0.2.Supposealsothatarandomsampleof20itemsisdrawnfromthepopulation.LetYdenotethenumberofdefectiveitemsinthesample,andconsideratestprocedureδsuchthatthecriticalregioncontainsalltheoutcomesforwhicheitherY≥7orY≤1.Determinethesizeofthetest.4.(TextbookSection8.1-4,Page300)SupposethatX1,...,Xnformarandomsamplefromanormaldistributionforwhichthemeanµisunknownandthevarianceis1.Supposealsothatµ0isacertainspecifiednumber,andthatthefollowinghypothesesaretobetested:H0:µ=µ0;1H1:µ̸=µ0.Finally,supposethatthesamplesizenis25,andconsideratestproceduresuchthatH0istobeacceptedif|Xn−µ0|1.Weshallconsidertestproceduresoftheform”rejectH0ifX≥c”.ForeachpossiblevaluexofX,findthep-valueifX=xisobserved.2
