Lab 12 The Central Limit Theorem

The sampling distribution of sample means is a distribution formed by means of all samples of size \(n\) repeatedly taken from a population. If a population has mean \(\mu\) and standard deviation \(\sigma\), and all possible samples of size \(n\) are taken from the population then sampling distribution satisfies the following properties \(\mu_{\bar{x}}=\mu\) and \(\sigma_{\bar{x}}=\sigma/\sqrt{n}\).

The Central Limit Theorem states that the sampling distribution of sample means approaches a normal distribution as the sample size gets larger no matter the population distribution’s shape. This is especially true for sample sizes over 30. The greater the sample size, the better the approximation.

If the population itself is normally distributed, then the sampling distribution of sample means is normally distributed for any sample size \(n\).

In this lab, you will simulate the sampling distribution by generating some of the samples of a specific size and comparing the results with the theoretical values for actual sampling distribution.