Statistics and Probability

5.(20pts) Let X1, ..., Xn be a random sample from N(u1, o?). Let Y1, ..., Yn be a random sample from N(#2, 02). Besides, X; is independent of Y; for all i, j. In addition, M1, H2, 07, 0? are unknown, but of = 03. (a) Please derive the 100(1-o)% confidence interval for 3/1 -4/2, using the pivotal quantity method. (Please include the derivation of the pivotal quantity, the proof of its distribution, and the derivation of the confidence interval for full credit.) (b) Please derive the test for Ho : 3/1 - 4/2 = 0 versus Ha : 3/1 - 4/2 > 0 at the significance level o using the pivotal quantity method (Please include the derivation of rejection region for full credit.) (c) Suppose we know that of = o for Ho : M1 = Ho versus Ha : /1 > Ho(/1 = Ha > Mo) at the significance level a, what condition should n(sample size) satisfy in order to guarantee that the power of the test will not less than 1 - B? (Please show the entire derivation to get full credits)