question is attached

1. (36 points) Your professor assigns one of two types of exams, kind exams and cruel exams. The score each student earns on the exam is a normal random variable that is independent of all other student's scores. All students are equally prepared for the exam, so they all have the same probability distribution of potential scores. a Kind exams: ,LL = 52 and a = 13.5. o Cruel exams: ,LL 2 48 and a = 13.5. There are 81 students in the class. After the exam the professor will tell you the average for the class, X: 81 X = i=1 Xi 81 and using that, you plan to do a hypothesis test to determine what kind of exam it was. The following will be your null and alternative hypotheses. 0 H0: \"=48 H1: ,u> 48 You will use a one-sided hypothesis test (a) How high would X need to be for you to have a pvalue less than .05? In other words, nd the cutoff where if X is greater than that cutoff you reject the null hypothesis, and if X is less than that cutoff you fail to reject. (b) You use a signicance level of 5%. If the test is kind, not cruel, what is the probability you make a type 2 error? (c) Suppose the class had more than just 81 students. How large would the class have to be for you to have 99% power in your hypothesis test when the exam is kind? (Assume a signicance level of 5% and a onesided test)