62. For a random sample of n individuals taking a licensing exam, let Xi  1 if the ith individual in the sample passes the exam and Xi  0 otherwise (i 

1, . . . , n).

a. With p denoting the proportion of all exam takers who pass, show that the most powerful test of H0: p  .5 versus Ha: p  .75 rejects H0 when .

b. If n  20 and you want a

.05 for the test described in (a), would you reject H0 if 15 of the 20 individuals in the sample pass the exam?

c. What is the power of the test you used in (b)

when p  .75 [i.e., what is p(.75)]?

d. Is the test derived in

(a) UMP for testing H0: p 

.5 versus Ha: p

.5? Explain your reasoning.

e. Graph the power function p(p) of the test for the hypotheses of

(d) when n  20 and a

.05.