7.3 Using the central limit theorem (modified r. bloom)  (Page 3/5)

Given a random variable (RV) with known mean μ and known variance σ 2 , we are sampling with size n and we are interested in two new RV - sample mean, X ˉ ,and sample sum, Σ X . If the size n of the sample is sufficiently large, then X ˉ ∼ N nμ σ 2 n and Σ X ∼ N nμ n σ 2 . In words, if the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. And even more, the mean of the sampling distribution will equal the population mean and mean of sampling sums will equal n times the population mean. The standard deviation of the distribution of the sample means, σ n , is called standard error of the mean.