The Central Limit Theorem suggests that for any population distribution, the distribution of the sample mean becomes normal as the sample size increases.

The Central Limit Theorem suggests that for any population distribution, the distribution of the sample mean becomes normal as the sample size increases.

• A. True
• B. False
• C. Only if the population is normal
• D. Only if the variance is finite

Answer: (A)

Central limit theorem says that if you average a large number of independent observations from any population with finite mean and finite variance, the distribution of that average becomes approximately normal. The mean of the sample average is the population mean μ, and its spread is σ/√n, which shrinks as n grows. So, regardless of the original shape, the sampling distribution of the mean tends toward a normal distribution as the sample size increases, making the normal model a good approximation for large samples. The statement is true under the usual conditions (finite variance); if variance were infinite, the normal approximation isn’t guaranteed. It’s also worth noting that even if the population is already normal, the sample mean is normal for any sample size, but the theorem extends normality to non-normal populations as n grows.

Central limit theorem says that if you average a large number of independent observations from any population with finite mean and finite variance, the distribution of that average becomes approximately normal. The mean of the sample average is the population mean μ, and its spread is σ/√n, which shrinks as n grows. So, regardless of the original shape, the sampling distribution of the mean tends toward a normal distribution as the sample size increases, making the normal model a good approximation for large samples. The statement is true under the usual conditions (finite variance); if variance were infinite, the normal approximation isn’t guaranteed. It’s also worth noting that even if the population is already normal, the sample mean is normal for any sample size, but the theorem extends normality to non-normal populations as n grows.