Two-Tailed Hypothesis Tests

In a two-tailed hypothesis test, the alternative hypothesis has the form $H_a: \mu \neq \mu_0$. We reject $H_0$ when the sample mean lies significantly above $\mu_0$ OR significantly below it, so the rejection region consists of two pieces -- one in each tail of the sampling distribution.

Because $\alpha$ is the total probability of rejecting $H_0$ when it is true, that probability must be split equally between the two tails: $\alpha/2$ in each tail.

The critical values for a two-tailed z-test are therefore

$$\pm z_{\alpha/2},$$

where $z_{\alpha/2}$ is the value satisfying $P(Z > z_{\alpha/2}) = \alpha/2$.

For example, at $\alpha = 0.05$, we have $\alpha/2 = 0.025$, giving $z_{\alpha/2} = z_{0.025} = 1.96$. The critical values are $\pm 1.96$, and the rejection region is

$$z < -1.96 \quad \text{or} \quad z > 1.96.$$