2. Let Y1,Y2,...,Yn be i.i.d. from a normal distribution with mean and variance 2.

(a) If both and 2 are unknown, explain why you cannot obtain a uniformly most powerful rejection region for the one-sided hypothesis tests for .

(b) If both and 2 are unknown, derive a 90% confidence interval for the parameter . Now suppose = 0 and 2 is unknown.

(c) Find the uniformly most powerful rejection region for the hypothesis test H0 : 2 = 20

versus Ha : 2 > 20.

(d) If 20 = 0.5, find the constant k so that the hypothesis test in part (c) has a Type I error rate of = 0.01.

(e) Derive an exact 95% confidence interval for the parameter 2.