5.71 Multiple-choice tests. Here is a simple probability model for multiple-choice tests. Suppose that each student has probability p of correctly answering a question chosen at random from a universe of possible questions.

(A strong student has a higher p than a weak student.)

The correctness of an answer to a question is independent of the correctness of answers to other questions.

Jodi is a good student for whom p  0.88.

(a) Use the Normal approximation to find the probability that Jodi scores 85% or lower on a 100-question test.

(b) If the test contains 250 questions, what is the probability that Jodi will score 85% or lower?

(c) How many questions must the test contain in order to reduce the standard deviation of Jodi’s proportion of correct answers to half its value for a 100-item test?

(d) Laura is a weaker student for whom p  0.72. Does the answer you gave in part

(c) for the standard deviation of Jodi’s score apply to Laura’s standard deviation also?