3. (25 points) Suppose we randomly select 5 students from STAT3025. Let X1, ..., X5 denote their grades on the first quiz, and Y1, ..., Y5 denote the grades on the second quiz. Assume Xi N(41,01) and Y; ~ N(M2, o). The students finished their quizzes independently. Subject 1 2 3 4 5 X 71 60 73 73 75 Y 88 86 71 82 72 (a) (4 points) Let be an estimate of M1. What is your value of 1 [Determine estimate using either MOM or MLE estimator]? (b) (5 points) Using your result in part(a), construct the test at a = - 0.05 [Use critical values that are necessary from page 1] Ho : M1 = 67 VS. Ha: Mi 67 (c) (6 points) Construct a 95% two-sided confidence interval for M1. Show that your confidence interval agrees with your conclusion in part (b). (d) (10 points ) Is there any evidence that the students did better in the second exam? Use a 0.05 [Hint: Run a hypothesis test; Use critical values that are necessary from page 1; In this problem, doing better implies running a hypothesis test using alternate hypothesis that Mid = M2 Mi > 0]