Assuming that the Central Limit Theorem will apply, what is the probability that a randomly selected sample mean will be within one standard deviation of the population mean?
A) 33%
B) 50%
C) 68%
D) 95%

Question

Quizplus · Accepted Answer

The Central Limit Theorem states that the distribution of sample means will be approximately normal, regardless of the shape of the population, as long as the sample size is sufficiently large. Furthermore, the mean of the distribution of sample means will be equal to the population mean, and the standard deviation of the distribution of sample means will be equal to the standard error of the mean (which is equal to the population standard deviation divided by the square root of the sample size). Since we are told to assume that the Central Limit Theorem will apply, we can use the empirical rule (also known as the 68-95-99.7 rule) to estimate the probability that a randomly selected sample mean will be within one standard deviation of the population mean. Specifically, we know that about 68% of sample means will fall within one standard error of the population mean (since one standard error is equal to one standard deviation divided by the square root of the sample size). Therefore, our answer is choice C, 68%.

Assuming That the Central Limit Theorem Will Apply, What Is