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 Professor Elderman's class of 36 has a class average below 78?
A) 0.1587
B) 0.5675
C) 0.8413
D) Cannot be determined.

Question

Quizplus · Accepted Answer

To find the probability that the class average is below 78, we use the Central Limit Theorem, which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough (n ≥ 30 is a common rule of thumb), regardless of the shape of the population distribution. Given that the sample size is 36 (which is greater than 30), we can proceed with this assumption.The mean of the sample means (μx̄) is equal to the population mean (μ), which is 76. The standard deviation of the sample means (σx̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (n), which is 12 / sqrt(36) = 12 / 6 = 2.We are looking for the probability that the sample mean is less than 78. We first calculate the z-score for a sample mean of 78:z = (X - μx̄) / σx̄ = (78 - 76) / 2 = 2 / 2 = 1.Looking up the z-score of 1 in the standard normal distribution table, or using a calculator, we find that the area to the left of z = 1 (which represents the probability that the sample mean is less than 78) is approximately 0.8413. This means there is an 84.13% chance that the class average will be below 78.

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