Hypothesis Tests for One Mean: Known Population Variance

In a hypothesis test for a single mean with known population variance, we compare a sample mean $\bar{x}$ to a hypothesized population mean $\mu_0$ using the test statistic

$$Z = \dfrac{\bar{x} - \mu_0}{\sigma / \sqrt{n}},$$

where $\sigma$ is the (known) population standard deviation and $n$ is the sample size. The denominator $\sigma/\sqrt{n}$ is the standard error of the sample mean.

Under the null hypothesis $H_0\!:\mu = \mu_0,$ the Central Limit Theorem guarantees that $Z$ is approximately standard normal for large $n$ (and exactly standard normal if the underlying population is itself normal).

For instance, suppose $\sigma = 10,$ $n = 25,$ and a sample produces $\bar{x} = 54,$ while $H_0$ asserts $\mu_0 = 50.$ Then

$$Z = \dfrac{54 - 50}{10 / \sqrt{25}} = \dfrac{4}{2} = 2.$$