Homework111.(TextbookSection8.3-1,Page322)SupposethatacertaindrugAwasadministeredtoeightpatientsselectedatrandom,andafterafixedtimeperiod,theconcentrationofthedrugincertainbodycellsofeachpatientwasmeasuredinappropriateunits.Supposethattheseconcentrationsfortheeightpatientswerefoundtobeasfollows:1.23,1.42,1.41,1.62,1.55,1.51,1.60,and1.76.SupposealsothataseconddrugBwasadministeredtosixdifferentpatientsselectedatrandom,andwhentheconcentrationofdrugBwasmeasuredinasimilarwayforthesesixpatients,theresultswereasfollows:1.76,1.41,1.87,1.49,1.67,and1.81.Assumingthatalltheobservationshaveanormaldistributionwithacommonunknownvariance,testthefollowinghypothesesatthelevelofsignificance0.10:Thenullhy-pothesesisthatthemeanconcentrationofdrugAamongallpatientsisatleastaslargeasthemeanconcentrationofdrugB.ThealternativehypothesesisthatthemeanconcentrationofdrugBislargerthanthatofdrugA.2.(TextbookSection8.4-7Page331)Considertwodifferentnormaldistributionsforwhichboththemeansµ1andµ2andthevariancesσ21andσ22areunknown,andsupposethatitisdesiredtotestthefollowinghypotheses:H0:σ21≤σ22,H1:σ21>σ22.Supposefurtherthatarandomsampleconsistingof16observationsforthefirstnormaldistributionyieldsthevalues�16i=1Xi=84and�16i=1X2i=563,andanindependentrandomsampleconsistingof10observationsfromthesecondnormaldistributionyieldsthevalue�10i=1Yi=18and�10i=1Y2i=72.(a)WhataretheM.L.E.’sofσ21andσ22?(b)IfanFtestiscarriedoutatthelevelofsignificance0.05,isthehypothesisH0acceptedorrejected?3.(TextbookSection9.1-4Page340)Accordingtoasimplegeneticprinciple,ifboththemotherandthefatherofachildhavegenotypeAa,thenthereisprobability1/4thatthechildwillhavegenotypeAA,theprobability1/2thatshewillhavegenotypeAa,1andprobability1/4thatshewillhavegenotypeaa.Inarandomsampleof24childrenhavingbothparentswithgenotypeAa,itisfoundthat10havegenotypeAA,10havegenotypeAa,andfourhavegenotypeaa.Investigatewhetherthesimplegeneticprincipleiscorrectbycarryingoutaχ2testofgoodness-of-fit.4.(TextbookSection9.2-2Page347)Atthefifthhockeygameoftheseasonatacertainarena,200peoplewereselectedatarandomandaskedhowmanyofthepreciousfourgamestheyhadattended.TheresultsaregiveninTable9.5.Testthehypothesisthatthese200observedvaluescanberegardedasarandomsamplefromabinomialdistribution;thatis,thereexitsanumberθ(0<θ<1)suchthattheprobabilitiesareasfollows:p0=(1−θ)4,p1=4θ(1−θ)3,p2=6θ(1−θ)2,p3=4θ3(1−θ),p4=θ4.NumberofgamespreviouslyattendedNumberofpeople0331672663154195.LettheresultofarandomexperimentbeclassifiedasoneofthemutuallyexclusiveandexhaustivewaysA1,A2,A3andalsoasoneofthemutuallyexclusiveandexhaustivewaysB1,B2,B3,B4.Twohundredindependenttrialsoftheexperimentresultinthefollowingdata:B1B2B3B4A11021156A211272113A36192724Test,atthe0.05significancelevel,thehypothesisofindependenceoftheAattributeandtheBattribute,namelyH0:Pr(AiBj)=Pr(Ai)Pr(Bj),i=1,2,3andj=1,2,3,4,againstthealternativeofdependence.2
