Statistical Inference III

QUESTION 1 [11] Suppose that X1, X2, X3, ..., X,1 is a random sample om a distribution with probability density function: ten'9} if): 5 5', ftxl} = t) otherwise. Let '1"" = max{X1, X2, ..:, X}. Given: The cumulative distribution function of 1" is: Eli\"l if r s 6, GUIE} = 1 if r 2 6*. [3) Find :3 r i}. in terms ofn, which satises 0.95 = P [6 - c 5 T S E}. {5} lb} Use part {a} to determine a 95% condence interval for :9 in terms :1. (5} QUESTION 2 [25] Refer to QUESTION 1. [3) Consider testing Hg : = -2 against H1 : = 1 at the 0.1 level of signicance using a test that rejects Hg if I}, 2: c. {I} What is c in terms of n? (5} {ll} What is the power of the test, in terms of n, at H = 1'? {T} {b} Suppose that another test of the hypotheses in part (a) rejects Hg at the [1.1 level of signicance if I}, 5 c. {i} What is r.- in terms of n? (5} {ll} What is the power of the test, in terms of n. at 5' = 1? {7} {III} Which of this test and the test in part {a} should be preferred and why? (2}