Which statement summarizes the Central Limit Theorem?

Which statement summarizes the Central Limit Theorem?

• A. The population distribution becomes normal as sample size increases.
• B. The sampling distribution of the sample mean becomes approximately normal for large n, regardless of the population shape.
• C. The sampling distribution of the sample mean is always exactly normal, regardless of n.
• D. It requires the population variance to be known to apply.

Answer: (B)

The central idea is that the distribution of the sample mean becomes normal as the sample size grows, regardless of the population’s shape, as long as the population has finite variance. This means that if you took many samples of size n and computed their means, those means would form a roughly bell-shaped distribution with mean equal to the population mean and variance equal to sigma^2/n. As n increases, that sampling distribution becomes increasingly close to normal. That’s why the statement describing an approximately normal sampling distribution for large n, no matter the population shape, is the best choice. It captures the essence of the Central Limit Theorem: the mean of samples tends toward normality with large n. It’s not the population itself that becomes normal as you sample more, so the first idea isn’t correct. And the distribution of the sample mean isn’t always exactly normal for finite n—only approximately normal for large n (and exactly normal if the population is normal to begin with). The theorem also doesn’t require knowing the population variance to apply, only finite variance.

The central idea is that the distribution of the sample mean becomes normal as the sample size grows, regardless of the population’s shape, as long as the population has finite variance. This means that if you took many samples of size n and computed their means, those means would form a roughly bell-shaped distribution with mean equal to the population mean and variance equal to sigma^2/n. As n increases, that sampling distribution becomes increasingly close to normal.

That’s why the statement describing an approximately normal sampling distribution for large n, no matter the population shape, is the best choice. It captures the essence of the Central Limit Theorem: the mean of samples tends toward normality with large n.

It’s not the population itself that becomes normal as you sample more, so the first idea isn’t correct. And the distribution of the sample mean isn’t always exactly normal for finite n—only approximately normal for large n (and exactly normal if the population is normal to begin with). The theorem also doesn’t require knowing the population variance to apply, only finite variance.