Most binomial probability calculations required in these exercises can be done by using Table C or the Normal approximation. Your instructor may request that you use the binomial probability formula or software. In exercises requiring the Normal approximation, you should use the continuity correction if you studied that topic.

Question 5.33

5.33 Multiple-choice tests.

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

  1. Use the Normal approximation to find the probability that Emily scores 85% or lower on a 100-question test.
  2. If the test contains 250 questions, what is the probability that Emily will score 85% or lower?
  3. 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?
  4. Diane is a weaker student for whom . Does the answer you gave in part (c) for the standard deviation of Emily’s score apply to Diane’s standard deviation also?

5.33

(a) 0.1788. (b) 0.0721. (c) . (d.) Yes.