Statistical Inference for Continuous Variables

Abstract

Suppose, hypothetically, we chose a random sample of n subjects from some infinitely large population of known mean μ and standard deviation σ, determined the mean ($\overline x$) of that sample, replaced the same subjects back into the source population, then chose another random sample of the same size n, and repeated this process over and over again. What distribution would the repeated sample $\overline x$’s have? It turns out that if n is large enough, then the $\overline x$’s form a normal distribution, regardless of the distribution of the source population. The mean of this normal sampling distribution of $\overline x$’s is μ, the population mean; its standard deviation (called the standard error of the mean, or SEM) is $\frac{\sigma }{{\sqrt n }}$.