A simple random sample of size n=10 is obtained from a population that is normally distributed with a mean of 40 and a standard deviation of 3. Is the sampling distribution normally distributed? Why?

Question

A. Yes, the sampling distribution is normally distributed because the population is normally distributed.

B. Yes, the sampling distribution is normally distributed because the population mean is greater than 30.

C. No, the sampling distribution is not normally distributed because the population is not normally distributed.

D. No, the sampling distribution is not normally distributed because the sample size is less than 30.

GPT-4o mini · Answer

The correct answer is:

A. Yes, the sampling distribution is normally distributed because the population is normally distributed.

Explanation: According to the Central Limit Theorem, if the population from which the samples are drawn is normally distributed, then the sampling distribution of the sample mean will also be normally distributed, regardless of the sample size. Since the given population is normally distributed with a mean of 40 and a standard deviation of 3, the sampling distribution of the sample mean for a sample size of 10 will also be normally distributed.