7.40 A multiple-choice exam consists of 50 questions.

Each question has five choices, of which only one is correct. Suppose that the total score on the exam is computed as y  x1 2 1

4 x2 where x1 5 number of correct responses and x2 5 number of incorrect responses. (Calculating a total score by subtracting a term based on the number of incorrect responses is known as a correction for guessing and is designed to discourage test takers from choosing answers at random.)

a. It can be shown that if a totally unprepared student answers all 50 questions by just selecting one of the five answers at random, then mx1  10 and mx2  40. What is the mean value of the total score, y? Does this surprise you? Explain. (Hint: See Example 7.16.)

b. Explain why it is unreasonable to use the formulas given in this section to compute the variance or standard deviation of y.