Which theorem states that the sampling distribution of the mean approaches normality as n increases?

Which theorem states that the sampling distribution of the mean approaches normality as n increases?

• A. Central Limit Theorem
• B. Normal Distribution
• C. Probability
• D. Unit Normal Table

Answer: (A)

The central limit theorem explains why the sampling distribution of the mean becomes normal as the sample size grows. When you take many independent samples of size n from a population and compute each sample mean, the distribution of those means tends to look bell-shaped, even if the population itself isn’t normally distributed. As n increases, that distribution centers around the true population mean and its spread shrinks like sigma divided by the square root of n, where sigma is the population standard deviation. This averaging effect smooths out irregularities in the population’s shape, which is why inference about means often uses normal-appearing approximations for large samples. The other terms refer to the normal distribution itself, the broader idea of probability, or a reference table, but they don’t specify the principle that the mean’s sampling distribution approaches normality with larger samples.

The central limit theorem explains why the sampling distribution of the mean becomes normal as the sample size grows. When you take many independent samples of size n from a population and compute each sample mean, the distribution of those means tends to look bell-shaped, even if the population itself isn’t normally distributed. As n increases, that distribution centers around the true population mean and its spread shrinks like sigma divided by the square root of n, where sigma is the population standard deviation. This averaging effect smooths out irregularities in the population’s shape, which is why inference about means often uses normal-appearing approximations for large samples. The other terms refer to the normal distribution itself, the broader idea of probability, or a reference table, but they don’t specify the principle that the mean’s sampling distribution approaches normality with larger samples.