Exercise 27 (#6.49). Let (Xi1, ..., Xini ), i = 1, 2, be two independent random samples from N(i, 2), respectively, where ni 2 and is and are unknown. Show that a UMPU test of size for H0 : 1 = 2 versus H1 : 1 = 2 rejects H0 when |t(X)| > tn1+n21,/2, where t(X) = (X2 X1) ;2 n1 1 + n1 2 [(n1 1)S2 1 + (n2 1)S2 2 ]/(n1 + n2 2), Xi and S2 i are the sample mean and variance based on Xi1, ..., Xini , i = 1, 2, and tn1+n21, is the (1 )th quantile of the t-distribution tn1+n21. Derive the power function of this test.