A professor is studying the performance of various student groups in his class. Let the random variable

Question:

A professor is studying the performance of various student groups in his class. Let the random variable X represent a student’s final score in the class and define the following conditioning events:
• F = {student is a freshman}
• So = {student is a sophomore}
• J = {student is a junior}
• Se = {student is a senior}
• M = {student is a male}
•F = {student is a female}
(a) Suppose the average score among males is E[X| M] = 73.2 and the average score among females is E[X| F] = 75.8 . If the overall class average score is E[X] = 74.6, what percentage of the class is female?
(b) Suppose the conditional average scores by class are E[X| F] = 65.8 , E[ X| So] = 71.2 , E[X| J] = 75.4 , E[ X| Se] = 79.1 . If the overall class average score is E[X] = 72.4, what can we say about the percentage of freshmen, sophomore, juniors and seniors in the class?
(c) Given the class statistics in part (b). If it is known that there are 10 freshmen, 12 sophomores, and 9 juniors in the class, how many seniors are in the class?