Professor Elderman has given the same multiple-choice final exam in his Principles of Microeconomics class for many years. After examining his records from the past 10 years, he finds that the scores have a mean of 76 and a standard deviation of 12. What is the probability that a class of 36 students will have an average greater than 70 on Professor Elderman's final exam?

Question

Quizplus · Accepted Answer

To find the probability that a class of 36 students will have an average score greater than 70, we use the formula for the standard error of the mean: $ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} $, where $ \sigma $ is the population standard deviation and $ n $ is the sample size. Here, $ \sigma = 12 $ and $ n = 36 $, so $ \sigma_{\bar{x}} = \frac{12}{\sqrt{36}} = 2 $. We then calculate the z-score for a mean of 70: $ z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} = \frac{70 - 76}{2} = -3 $. Using the z-table, a z-score of -3 corresponds to a probability of 0.00135, which is the probability of having a mean of 70 or lower. Therefore, the probability of having a mean greater than 70 is 1 - 0.00135 = 0.99865, which rounds to 0.9986.

Professor Elderman Has Given the Same Multiple-Choice Final Exam in His