A First Course in Statistics, 12th Edition

6.4 Test of Hypothesis about a Population Mean: Normal (z) Statistic

When testing a hypothesis about a population mean $\mu$, the test statistic we use will depend on whether the sample size n is large (say, $n\ge 30$) or small and whether we know the value of the population standard deviation, $\sigma$. In this section, we consider the large-sample case.

Because the sample size is large, the Central Limit Theorem guarantees that the sampling distribution of $\bar{x}$ is approximately normal. Consequently, the test statistic for a test based on large samples will be based on the normal z-statistic. Although the z-statistic requires that we know the true population standard deviation $\sigma$, this is rarely, if ever, the case. However, we established in Chapter 5 that ...