Which theorem states that as the sample size increases, the distribution of the sample means tends toward normal, regardless of the population distribution?

Which theorem states that as the sample size increases, the distribution of the sample means tends toward normal, regardless of the population distribution?

• A. Central limit theorem
• B. Law of Large Numbers
• C. Chebyshev's inequality
• D. Bayes' theorem

Answer: (A)

The main idea here is the Central Limit Theorem. It says that if you take a large number of independent, identically distributed random samples from any population with finite mean and variance, and compute their means, the distribution of those sample means becomes approximately normal as the sample size grows. The normal shape emerges regardless of how the population is distributed, and the spread of the distribution of the sample mean decreases with larger samples (specifically, its standard deviation is the population standard deviation divided by the square root of the sample size). This is the best answer because it directly describes why the means of many samples tend toward a normal distribution when the sample size increases, which is exactly the statement in the question. The other options describe different ideas: the Law of Large Numbers is about the sample mean converging to the true population mean, not about the shape of its distribution; Chebyshev's inequality provides a general bound on deviations but not the normal shape; Bayes' theorem deals with updating probabilities with new evidence rather than sampling distributions.

The main idea here is the Central Limit Theorem. It says that if you take a large number of independent, identically distributed random samples from any population with finite mean and variance, and compute their means, the distribution of those sample means becomes approximately normal as the sample size grows. The normal shape emerges regardless of how the population is distributed, and the spread of the distribution of the sample mean decreases with larger samples (specifically, its standard deviation is the population standard deviation divided by the square root of the sample size).

This is the best answer because it directly describes why the means of many samples tend toward a normal distribution when the sample size increases, which is exactly the statement in the question. The other options describe different ideas: the Law of Large Numbers is about the sample mean converging to the true population mean, not about the shape of its distribution; Chebyshev's inequality provides a general bound on deviations but not the normal shape; Bayes' theorem deals with updating probabilities with new evidence rather than sampling distributions.