Question

Which one of the following statements is true?

A. The Central Limit Theorem states that the sampling distribution of the sample mean, y , is approximately Normal for large n only if the distribution of the population is normal.

B. The Central Limit Theorem states that the sampling distribution of the sample mean, y , is approximately Normal for small n only if the distribution of the population is normal.

C. The Central Limit Theorem states that the sampling distribution of the sample mean, y , is approximately Normal for small n regardless of the distribution of the population.

D. The Central Limit Theorem states that the sampling distribution of the sample mean, y , is approximately Normal for large n regardless of the distribution of the population.

Please Explain and show work if possible. Thank you!

Answer 1

First of all, the central limit theorem assumes the sample size to be large enough to make any conclusions. This means we can reject option B and C as these options are dealing with small sample sizes.

Now, we have to pick the right option out of A and D only

We know that the central limit theorem suggests that as the sample size increases, or it becomes large enough then the sampling distribution of the mean is said to be approximately normally distributed, irrespective of the individual sample distribution in the population.

This means there is no necessity for the population to be normally distributed while applying the central limit theorem, So option A can also be rejected.

Thus, only option D is the correct answer