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. Emily is a good student for whom p 5 0.88.

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

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

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

(d) Diane is a weaker student for whom p 5 0.72.

Does the answer you gave in part

(c) for the standard deviation of Emily’s score apply to Diane’s standard deviation also?