Problem 1.4: Testing a difference in means (30 points, 10 points each). Compare means. a. Consider two groups {X1, ..., Xm} and {Y1, ..., Yn}. We suppose that Xi ~i.id. F(M1, 1) and Yi ~i.i.d. F(12, 02). Additionally, we suppose that X and Y are independent samples. We are trying to estimate the difference in means, M1 - /2. What sample estimator should we use? b. Prove that your estimator is unbiased, and that it has a standard deviation of OT m n c. Now use your estimator to write a test statistic for the following test: Ho : M1 - H2 = Do H1 : M1 - 12 > Do Recall that the general version of a test statistic is T = estimator - assumed value in Ho s.d. of your estimator