65. (i) Let (XI' YI ) , .. . ,(Xn , Yn ) be a sample from the bivariate normal distribution (70), and let Sr = '£( X; - X)2, Sf = '£( Y; - y)2, Sl2 = '£( X; - X)( Y; - Y). There exists a UMP unbiased test for testing the hypothesis T/ a = t:J.. Its acceptance region is 1t:J.2 S2 - S21 I 2 C < , V( t:J.2 Sr + Sf)2 - 4t:J.2Sr2 - and the probability density of the test statistic is given by (84) when the hypothesis is true. (ii) Under the assumption T =

a, there exists a UMP unbiased test for testing '1/ = t with acceptance region iY - XV /Sf + Sf - 2Sl2 S C. On multiplication by a suitable constant the test statistic has Student's t-distribution with n - 1 degrees of freedom when '1/ = t (Without the assumption T =

a, this hypothesis is a special case of the one considered in Chapter 8, Example 2.) [(i): The transformation U = f:.X + Y, V = X - (l/t:J.)Y reduces the problem to that of testing that the correlation coefficient in a bivariate normal distribution is zero. (ii): Transform to new variables V; = Y; - X;, U; = Y; + X;.]