QUESTION 3 [29] Let X1, X2, ..., X20 be a random sample from a N(0, 2) distribution. It is desired to construct a 95% confidence interval for 2 and to test Ho : 62 = 2 against H1 : 62 > 2 at the 0.05 level of significance. Note: The sample size is n = 20. 71 20 (a) Justify that the distribution of 2 = > is a x3, distribution? (3) 1=1 (b) Find a and b (a b). (4 20 20 (c) Use parts (a) and (b) to show that ( 0.0293 ) x?, 0.1043 ) x? is a 95% confidence 1=1 i=1 interval estimator for o2. Hint: What is P(a s Q s b)? 6) (d) Show that the uniformly most powerful test for Ho : 62 = 2 against H1 : 62 > 2, at the 20 0.05 level of significance, rejects Ho if Q = 2 2 31.41. Note: The sample size is n = 20. (16) TOTAL MARKS: 85