The dean of admissions in a large university has determined that the scores of the freshman class in a mathematics test are normally distributed with a mean of 82 and a standard deviation of 8. Based on this information, what is the probability that the mean score of a sample of 64 students is at least 80?
A) .0987
B) .4772
C) .5987
D) .7885
E) .9772

Question

Quizplus · Accepted Answer

Given that the scores are normally distributed with a mean (μ) of 82 and a standard deviation (σ) of 8, and the sample size (n) is 64, we can use the Central Limit Theorem to find the probability that the sample mean is at least 80. The standard error (SE) of the mean is σ/√n = 8/√64 = 1. The Z-score for a sample mean of 80 is (80 - 82)/1 = -2. Using the Z-table, a Z-score of -2 corresponds to a left-tail probability of 0.0228. Therefore, the probability of the sample mean being at least 80 is 1 - 0.0228 = 0.9772.

The Dean of Admissions in a Large University Has Determined