A company claims that it has a magic spell that improves the test scores of students who are doing badly in school. It has large amounts of data that show that, when it casts its spell on students who score in the bottom 1% on a standardized test, their scores improve significantly (on average) when they take a similar test a second time. In this question, we’ll evaluate whether this necessarily means that their spell works. We’ll do this by seeing if we can explain this data assuming the spell doesn’t do anything.1 Suppose that a student’s test score is a combination of skill and random chance. The student’s skill won’t change when he takes the test twice, but the random part of his score will. Formally, let S, R1, and R2 be independent random variables such that:
S is uniform on (0,2).
Ri is uniform on (0,1].
R2 is uniform on (0,1).
If you pick a random student and give him two tests, his score on the first test will be given by Ti = S+R1, and his score on the second test will be given by T, =S + R2. (a) For a € (0,1), compute the probability that T S a. (We only care about students who did badly, so we won't bother dealing with the case when a > 1
a. Find the value of a such that PT Sa) = 0.01.
b. For a, b € (0,1), compute P R S T c. For a € (0, 1), use your answer to part (b) to compute the probability density function and the expected value of R1 given that Ti as.
d. Given that Tisa (for a € (0.1)), what is the expected value of 7 71? In particular, what is the expected change in a randomly chosen student's score, given that he scored in the bottom 1% on the first exam?
e. Briefly explain in words why the company's spell appears to improve students in the bottom 1% even though it doesn't do anything.