Large-Sample Z-Test for a Single Mean

When the sample size is large ($n \geq 30$, by the usual convention) and either the population standard deviation $\sigma$ is known or the sample standard deviation is used as a close approximation to it, a claim about the population mean is tested using the Z-test. The test statistic is:

where $\bar{x}$ is the sample mean, $\mu_0$ is the value of the population mean claimed under $H_0$, $\sigma$ is the population standard deviation and $n$ is the sample size. Notice that the denominator is exactly the standard error from earlier in this chapter — the Z-statistic simply measures how many standard errors the sample mean lies away from the claimed value.

The general steps of the test:

1. State $H_0$ and $H_1$ (and hence whether the test is one-tailed or two-tailed).
2. Choose the level of significance $\alpha$.
3. Compute the test statistic $Z$ from the sample.
4. Find the critical value from the standard normal table for the chosen $\alpha$ and tail-type.
5. Compare and conclude: reject $H_0$ if the calculated $|Z|$ (two-tailed) or $Z$ (one-tailed, with the correct sign) exceeds the critical value; otherwise do not reject $H_0$. …