4.(a) The random variable,Yis the number oftigers with the bacteria present.A rea-sonable model (given the available infor-mation) isY∼B(15, p).The hypothe-ses can be written:H0:p= 0.05 vsHA:p >0.05. Evidence against the nullis obtained for ’large’ observed values ofY. Using the basic definition of p-values,p-val=P(Y≥2). This can be writtenas 1-(P(Y= 0) +P(Y= 1)). Using thebinomial formula results inp-val= 0.171.This means that there is no meaningful ev-idence against the claim that the rate ofoccurrence of this bacteria is 0.05 or less.(b) The random variableYis the number ofcats with bacteria present.A reasonablemodel isY∼B(60, p).H0:p= 0.15 vsHA:p≥0.15. Sincenp(= 9) andn(1-p)(= 51) are both larger than 5, the nor-mal approx may be used.Let ˆp=Y/60.Then, underH0,ˆpNA∼N(.15,(.0461)2).α=P(ˆp≥0.25) =P(Z≥2.17) =.015.5.(a) The general form for a CI for the differencebetween two proportions is: ( ˆpA-ˆpB)±Zα/2*pA*(1-pA)nA+pB*(1-pB)nB.BecausepA= 0.4 andpB= 0.6, both expressionsof the formp(1-p) are the same.SinceZα/2affects all CIs equally (given the same1-α
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