Suppose that X1, ..., Xn is a random sample from a gamma distribution with parameters o and 3. (a) (3 points) If a = m, where m is a known integer, and S is unknown, find a pivotal quantity that has a X2, distribution. Use this pivotal quantity to derive a 100(1 - o)% confidence interval for B. (b) (3 points) If a = c, where c is a known constant but not an integer, and S is unknown, show that _ Xi/B is a pivotal quantity. Use this pivotal quantity to derive a 100(1 -a)% confidence interval for B. Problem 2 (3 points) Suppose S" is the sample variance based on a random sample of size n from a N(u, o2) with unknown mean / and unknown variance o2. (a) (1 point) Derive a 100(1 - @)% upper confidence bound for o2. (b) (1 point) Derive a 100(1 - @) % lower confidence bound for o2. (c) (1 point) Derive a 100(1 - 20)% confidence interval for o