Illuminating Statistical Analysis Using Scenarios and Simulations

27The t-Distribution

In Chapter 22, we used the population variance to calculate the standard error of a sample mean via . In practice, we rarely know the population variance and so in subsequent chapters we used the sample variance instead to calculate an estimate of the standard error of a sample mean via . But, as noted in Chapter 23, this introduces additional uncertainty, particularly when we have small sample sizes.

Let's look at the results of four random sampling simulations involving sample sizes of 4, 7, 60, and 1000. Each of the four simulations randomly samples from a population with population mean of 10 and population variance of 4 (there is nothing special about these values). Results are shown in Figures 27.1–27.4. The figure “a” histograms show the distributions of the sample variances, , for each of the four sample sizes. The corresponding figure “b” histograms show the distributions of the #SEs calculated via , which is the difference between a sample mean and the population mean rescaled by the standard error estimate. The horizontal axes end at −1.96 and +1.96 to ...