A normally distributed population has a mean of 40 and a standard deviation of 12. What does the central limit theorem say about the sampling distribution of the mean if samples of size 100 are drawn from this population.

The central limit theorem (CLT) states that as the sample size increases, the sampling distribution of the mean will approach a normal distribution, regardless of the shape of the population distribution.

To understand how the central limit theorem applies to this question, we need to consider the population distribution and the sample size. In this case, the population is normally distributed with a mean of 40 and a standard deviation of 12.

When samples of size 100 are drawn from this population, the central limit theorem tells us that the sampling distribution of the mean will also be normally distributed. The mean of the sampling distribution will be the same as the population mean (40), and the standard deviation of the sampling distribution (also known as the standard error) will be equal to the population standard deviation divided by the square root of the sample size.

Therefore, for this specific example, the sampling distribution of the mean, when drawing samples of size 100 from a normally distributed population with a mean of 40 and a standard deviation of 12, will have a mean of 40 and a standard deviation of 12 divided by the square root of 100, which is 1.2.

In summary, the central limit theorem states that the sampling distribution of the mean will be normally distributed as the sample size increases, regardless of the population distribution.