1. A coin is thrown independently twenty times to test the null hypothesis that the probability of heads is = against the alternative that the probability is not }. The test rejects if either ( heads or 20 heads are observed. (a) What is the significance level of the test? (b) If the true probability of heads is 0.2, what is the power of the test? 2. X has one of the two distributions in the following table. We observe one realization of X. I P(X = I| Ho) P(X = : H,) 0.2 0.1 0.3 0.4 0.3 0.1 TA 0.2 0.4 (a) Compute the likelihood ratio, A, for each possible value X, and order the r; from smallest to largest A. (b) What is the likelihood ratio test of the null hypothesis Ho against the alternative Hat level o = 0.2? (c) What is the power of the test in part (b)? 3. Suppose that X has a binomial distribution with n = 100. We wish to test the null hypothesis that p =0.5 against the alternative that p = 0.5. (a) Construct a Wald test with significance level 0.05. (b) Using probabilities from the binomial distribution, find the true significance level of your Wald test from part (a). 4. There is a theory that people can postpone their death until after an important event. To test the theory, Phillips and King (1988) collected data on deaths around the Jewish holiday Passover. Of 1919 deaths, 922 died the week before the holiday and 997 died the week after. Think of this as a binomial and test the null hypothesis that the success probability 0 = 1/2. Here "success" means people die after the holiday. Report and interpret the P-value. Also construct a confidence interval for 0. 5. Let X1,. .., Xe ~ Uniform(0, 0) and let Y = max( X1, ..., Xn). We want to test: Ho : 0 =1/2 versus H1 : 0 > 1/2. The Wald test is not appropriate since Y does not converge to a Normal. Suppose we decide to test this hypothesis by rejecting Ho when Y > c. (a) Find the power function. (b) What choice of c will make the size of the test 0.05? (e) In a sample of sixe n = 20 with Y =0.48, what is the p-value? What conclusion about Ho would you make? (d) In a sample of size n = 20 with Y =0.52, what is the p-value? What conclusion about Ho would you make