What does the Central Limit Theorem imply as sample sizes increase?

The Central Limit Theorem states that as the sample size increases, the distribution of the sample means will increasingly approximate a normal distribution, regardless of the original distribution of the population from which the samples are drawn, provided the samples are independent and identically distributed. This is significant because it allows for the application of statistical methods that rely on normality, making it possible to infer properties about the population mean even when the underlying population distribution is not normal.

This convergence to normality occurs because larger sample sizes tend to average out extreme values and random fluctuations in the data. As such, with a sufficiently large sample size, the sample means will form a distribution that is not only normal but also characterized by a mean equal to that of the population and a standard deviation that is proportional to the population standard deviation divided by the square root of the sample size. Therefore, statement B accurately captures the essence of the Central Limit Theorem.

The other statements do not accurately reflect the implications of the Central Limit Theorem. For instance, while variability may decrease as sample size increases—specifically in relation to the standard error of the mean—it does not necessarily decrease significantly for the overall population data. Additionally, not all samples will converge to the same mean; instead, their