Question

The Central Limit Theorem states that the sampling distribution of the sample mean is approximately normal under certain conditions. Which of the following is a necessary condition for the Central Limit Theorem to be used?

A. The population size must be large (e.g., at least 30).

B. The population from which we are sampling must be normally distributed.

C. The sample size must be large (e.g., at least 30)

D. The population from which we are sampling must not be normally distributed.

Answer 1

The Central Limit Theorem requires a sufficiently large sample size, generally at least 30, to approximate the sampling distribution of the sample mean as normal.

Step-by-step explanation:

The necessary condition for the Central Limit Theorem (CLT) to be applied is that the sample size must be large (typically n ≥ 30). While the CLT works better when the population distribution is normal, it is not a requirement. The crucial aspect is the sample size because as the sample size increases, the sampling distribution of the sample mean will become approximately normal regardless of the population's distribution.

This aligns with the law of large numbers, which suggests that the larger the sample size, the closer the sample mean will be to the population mean.

The necessary condition for the Central Limit Theorem to be used is C. The sample size must be large (e.g., at least 30). The Central Limit Theorem states that the sampling distribution of the sample mean will approach normality, regardless of the distribution of the population, if the sample size is sufficiently large.

This means that as the sample size increases, the shape of the distribution of the sample means becomes more and more like a normal distribution. Therefore answer is C. The sample size must be large (e.g., at least 30).