Suppose that the test score of a student taking the final of a probability course is a random variable with mean 77.

(a) Give an upper bound for the probability that a student's test score will exceed 87.
$P\lbrace \mbox{score} > 87 \rbrace \le$

(b) Suppose that we know, in addition, that the variance of students' test scores on the final is 26. What can you say about the probability that a student will score between 67 and 87 (do not use the central limit theorem)?
$P\lbrace 67\le \mbox{score}\le 87\rbrace$ ? <= >= =

(c) How many students would have to take the final to ensure with a probability of at least 0.85 that the class average would be within 5 of 77 (do not use the central limit theorem)?
$n =$

(d) If you use the central limit theorem in (c), what is your estimate for the number of students needed?
$n =$