Statistics and Probability

Let X be a normal random variable with unknown mean u and unknown variance 02. . The null hypothesis is that #0 = 1 and 03 = 1; . The alternative hypothesis is that #1 = 2 and of = 4. Let X1, . . .Xn be 71 = 20 independent samples of X , and let R Z := Zin- = X1+2X2+...+an. i=1 (a) Compute the expectation and variance of Z , rst under the null hypothesis, then under the alternative hypothesis. Let C be the critical region 0 := {(X1,...,Xn) | Z 2 In}. Find the value 16 so that the type I error rate of this critical region is a = 0.1. Compute the type II error rate for the test in part (b)