7.1 The central limit theorem for sample means (averages)  (Page 16/18)

Given a random variable (RV) with known mean μ and known standard deviation, σ , we are sampling with size n , and we are interested in two new RVs: the sample mean, X ¯ , and the sample sum, ΣΧ . If the size ( n ) of the sample is sufficiently large, then X ¯ ~ N ( μ , σ n ) and ΣΧ ~ N ( nμ , ( n )( σ )). 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 distributions regardless of the shape of the population. The mean of the sample means will equal the population mean, and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means, σ n , is called the standard error of the mean.