The Central Limit Theorem only applies to populations that are normally distributed.

The Central Limit Theorem only applies to populations that are normally distributed.

• A. True
• B. False
• C. Depends on sample size
• D. Depends on population variance

Answer: (B)

The central idea is that the distribution of the sample mean becomes approximately normal as you take larger samples, regardless of the population’s shape, as long as the population has a finite mean and finite variance. This means the population does not have to be normally distributed for the normal approximation to hold. If you repeatedly draw samples of size n from any distribution with finite variance and compute their means, the spread of those sample means forms a distribution that converges to a normal distribution with mean equal to the population mean and variance equal to the population variance divided by n, as n grows larger. The population could be skewed or U-shaped, and you’d still get a near-normal sampling distribution for sufficiently large n. The conditions to keep in mind are independence (or weak dependence), identical distribution, and finite variance. For small n, the sampling distribution of the mean may not look normal, especially if the population is far from normal or has heavy tails. But the theorem guarantees normality in the limit as n grows, not only for populations that start out normal.

The central idea is that the distribution of the sample mean becomes approximately normal as you take larger samples, regardless of the population’s shape, as long as the population has a finite mean and finite variance. This means the population does not have to be normally distributed for the normal approximation to hold.

If you repeatedly draw samples of size n from any distribution with finite variance and compute their means, the spread of those sample means forms a distribution that converges to a normal distribution with mean equal to the population mean and variance equal to the population variance divided by n, as n grows larger. The population could be skewed or U-shaped, and you’d still get a near-normal sampling distribution for sufficiently large n. The conditions to keep in mind are independence (or weak dependence), identical distribution, and finite variance.

For small n, the sampling distribution of the mean may not look normal, especially if the population is far from normal or has heavy tails. But the theorem guarantees normality in the limit as n grows, not only for populations that start out normal.