Question 3: (7 Total Marks) Let X1, ,Xloo be iid Bern(p) and assume we observe 2113? X1- = 99 (Le, a single failure). a) Construct a 95% CLT-based condence interval for p. Briey comment on any issues with the endpoints you get for your interval. [2 marks] Recall from lectures that for estimating p from this probability model, we have 155mg = (1 / n) 2&1 X i (i.e., the usual sample proportion) and Fisher information 109) = n/{p(1 19)}. b) Suppose \"((19) = ln(1 p). Construct a 95% MLEbased condence interval for \"((39). Also, discuss how you could use your interval for 109) to derive an interval for p different to the one from (a) and briey comment on its benets and drawbacks. [2 marks] Page 3 of 4 Question 3 (Continued): Instead of a xed sample of 100 trials, suppose we observe trials until the second success occurs and let N ~ NB (2, p) be the number of failures before the second success. In this case, the likelihood is: L(p) = (N + 1)p2(1 p)\" => {(39) = ln(N + 1) + 2 ln(p) + N ln(1 p) c) Assume we observe N = 198. Calculate the pvalue of a test of the null hypothesis H0: p = 0.5 versus the two-sided alternative H A: p at 0.5. [3 marks]