3.3 One-sample means with the $t$ distribution

The motivation in Chapter 2.6 for requiring a large sample was two-fold. First, a large sample ensures that the sampling distribution of $\overline{x}$ is nearly normal. We will see in Section 3.3.1 that if the population data are nearly normal, then $\overline{x}$ is also nearly normal regardless of the sample size. The second motivation for a large sample was that we get a better estimate of the standard error when using a large sample. The standard error estimate will not generally be accurate for smaller sample sizes, and this motivates the introduction of the $t$ distribution, which we introduce in Section 3.3.2.

We will see that the $t$ distribution is a helpful substitute for the normal distribution when we model a sample mean $\overline{x}$ that comes from a small sample. While we emphasize the use of the $t$ distribution for small samples, this distribution may also be used for means from large samples.