498 Optimal Tests of Hypotheses(d)Determine the smallest value ofnso the power exceeds 0.80 to detectμ= 53.
Hint:Modify the R functiontpowerg.Rso it returns the power for a specified
alternative.8.3.6. The effect that a certain drug (Drug A) has on increasing blood pressure
is a major concern. It is thought that a modification of the drug (Drug B) will
lessen the increase in blood pressure. LetμAandμBbe the true mean increases
in blood pressure due to Drug A and B, respectively. The hypotheses of interest
areH 0 :μA=μB=0versusH 1 :μA>μB=0. Thetwo-samplet-test statistic
discussed in Example 8.3.3 is to be used to conduct the analysis. The nominal level
is set atα=0.05 For the experimental design assume that the sample sizes are
the same; i.e.,m=n. Also, based on data from Drug A,σ= 30 seems to be a
reasonable selection for the common standard deviation. Determine the common
sample size, so that the difference in meansμA−μB= 12 has an 80% detection rate.
Suppose when the experiment is over, due to patients dropping out, the sample sizes
for Drugs A and B are respectivelyn=72andm= 68. What was the actual power
of the experiment to detect the difference of 12?
8.3.7.Show that the likelihood ratio principle leads to the same test when testing
a simple hypothesisH 0 against an alternative simple hypothesisH 1 ,asthatgiven
by the Neyman–Pearson theorem. Note that there are only two points in Ω.
8.3.8.LetX 1 ,X 2 ,...,Xnbe a random sample from the normal distributionN(θ,1).
Show that the likelihood ratio principle for testingH 0 :θ=θ′,whereθ′is specified,
againstH 1 :θ =θ′leads to the inequality|x−θ′|≥c.
(a)Is this a uniformly most powerful test ofH 0 againstH 1?(b)Is this a uniformly most powerful unbiased test ofH 0 againstH 1?8.3.9.LetX 1 ,X 2 ,...,Xnbe iidN(θ 1 ,θ 2 ). Show that the likelihood ratio principle
for testingH 0 :θ 2 =θ′ 2 specified, andθ 1 unspecified, againstH 1 :θ 2
=θ′ 2 ,θ 1
unspecified, leads to a test that rejects when
∑n
1 (xi−x)(^2) ≤c 1 or∑n
1 (xi−x)
(^2) ≥c 2 ,
wherec 1 <c 2 are selected appropriately.
8.3.10.For the situation discussed in Example 8.3.5, derive the power function for
the likelihood ratio test statistic given in expression (8.3.9).
8.3.11.LetX 1 ,...,XnandY 1 ,...,Ymbe independent random samples from the
distributionsN(θ 1 ,θ 3 )andN(θ 2 ,θ 4 ), respectively.
(a)Show that the likelihood ratio for testingH 0 :θ 1 =θ 2 ,θ 3 =θ 4 against all
alternatives is given by
[n
∑
1
(xi−x)^2 /n
]n/ 2 [m
∑
1
(yi−y)^2 /m
]m/ 2
{[n
∑
1
(xi−u)^2 +
∑m
1
(yi−u)^2
]/
(m+n)
}(n+m)/ 2 ,
whereu=(nx+my)/(n+m).