67. Suppose that each of n randomly selected individuals is classi ed according to his/her genotype with respect to a particular characteristic and that the three possible genotypes are AA, Aa, and aa with long-run proportions (probabilities) u2, 2u(1  u), and (1  u)2, respectively (0  u  1). It is then straightforward to show that the likelihood is where x1, x2, and x3 are the number of individuals in the sample who have the AA, Aa, and aa genotypes respectively. Show that the most powerful test for testingH0: u.5 versusHa: u.8 rejects the null hypothesis when 2x1  x2 

c. Is this test UMP for the alternative Ha: u .5? Explain. Note: The fact that the joint distribution of X1, X2, and X3 is multinomial can be used to obtain the value of c that yields a test with any desired signi cance level when n is large.