Exercise 31 (#6.54). Let (X1, ..., Xn) be a random sample from a bi- variate normal distribution with unknown means Hi and M2, variances o and ca, and correlation coefficient p. Let Xij be the jth component of Xi, j = 1,2, X; and Sy be the sample mean and variance based on X1j, ..., Xnj, and V = Vn 2R/V1 R2, where n 1 R= (Xi1 - 71)(X;2 - ) SiS2(n-1) i=1 is the sample correlation coefficient. Show that the UMPU test of size a for Ho :P 0 rejects Ho when V > tn-2,a and the UMPU test of size a for Ho : p = 0 versus Hj :P # O rejects Ho when |V] > tn-2,0/2, where tn-2,a is the (1 a)th quantile of the t-distribution tn-2. TT