H1 : M = 6 given that one draws 3 balls from the box without replacement. The null hypothesis is rejected if the sample contains 2 or 3 white balls. Otherwise, it is accepted. Find the probability of Type I error and the power of the test. 2. Let X1, . . ., Xn be iid from the Beta distribution with the pdf Pe(x) - 0x -, 00 (a) Find the most powerful test of level a for testing Ho : 0 =1 vs. H1 : 0 = 3. (b) Define T = -2 27_, log Xi. Show that T ~ Xzn- (Hint: First, try to show that if X follows a continuous distribution with X ~ Fx(x), then F(X) ~ Unif(0, 1)) (c) Let n = 10 and T = 10.8. What is your decision? Assume a = 0.05. 3. Let X be a random variable with the following pdf: fe(x) = 20x + 2(1 -0)(1-x), 0 60. 5. X1 , . . ., Xn N(M, 02), both u and of are unknown Find the level o GLRT of the hypothesis test: Ho : o =go VS. H1 :0 #60