Which theorem states that larger samples will make the sample mean closer to the population mean?

The law of large numbers is the principle that larger samples will result in the sample mean being closer to the true population mean. This theorem indicates that as the sample size increases, the sample mean converges to the expected value (or population mean), increasing accuracy. This concept is foundational in statistics, as it supports the idea that to obtain reliable estimates of population parameters, collecting larger samples is beneficial.

While the Central Limit Theorem is also an important concept, it specifically addresses the distribution of sample means when you take an infinite number of samples from a population. It states that, regardless of the population's distribution, the distribution of the sample means will approach a normal distribution as the sample size increases. However, the question specifically asks about the relationship between sample size and the accuracy of the sample mean in representing the population mean, a fundamental aspect of the Law of Large Numbers.

Bayes' theorem focuses on the probability of an event based on prior knowledge or beliefs and does not pertain to sample sizes or means. Similarly, the variance theorem relates to the concept of variance in different settings and does not address the relationship between sample sizes and sample means. Thus, the law of large numbers is the most relevant theorem in this context.