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?
5.33 Multiple-choice tests. Here is a simple
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?
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