Which theorem states that the distribution of sample means approaches a normal distribution as the sample size increases?

The correct answer is the Central Limit Theorem, which is fundamental in statistics. This theorem states that when you take sufficiently large samples from a population, the distribution of the sample means will approach a normal distribution, regardless of the shape of the population distribution, as long as the population has a finite mean and variance. This principle is essential in hypothesis testing and confidence interval estimation, as it allows statisticians to make inferences about population parameters even when the underlying population distribution is unknown.

The significance of this theorem lies in its application: it provides a way to apply normal probability techniques to situations where they might not otherwise be applicable, especially when dealing with non-normally distributed data. As the sample size increases, the distribution of the sample means becomes tighter and more concentrated around the population mean, highlighting the reliability of sample results in estimating population parameters.

The Law of Large Numbers, while related, focuses on the convergence of the sample mean to the expected value (population mean) as the sample size increases rather than on the distribution shape. The Normal Distribution Theorem is not a standard term within statistical theory and does not capture the essence of the phenomenon described by the Central Limit Theorem. The Sampling Distribution Theorem is also not widely recognized as a formal theorem