A multiple-choice exam consists of 50 questions. Each question has five choices, of which only one is

Question:

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 calculated as

where x₁ = number of correct responses and x₂ = 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 ?x₁ = 10 and ?x_2?= 40. What is the mean value of the total score, y? Does this surprise you? Explain. 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.

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