Critical Regions for Right-Tailed Hypothesis Tests

A right-tailed hypothesis test has an alternative hypothesis of the form $H_a: \theta > \theta_0$. We reject $H_0$ only when the test statistic is unusually large — that is, when it lies far in the right tail of the null distribution.

For a right-tailed z-test at significance level $\alpha$, the critical value $z_\alpha$ is the number satisfying

$$P(Z \geq z_\alpha) = \alpha,$$

where $Z$ is a standard normal random variable. The critical region (or rejection region) is

$$\{\, z : z \geq z_\alpha \,\}.$$

The most common right-tailed critical values are:

$$\alpha = 0.10 \;\Longrightarrow\; z_\alpha = 1.282$$ $$\alpha = 0.05 \;\Longrightarrow\; z_\alpha = 1.645$$ $$\alpha = 0.025 \;\Longrightarrow\; z_\alpha = 1.960$$ $$\alpha = 0.01 \;\Longrightarrow\; z_\alpha = 2.326$$

Unlike the left-tailed case, the critical region of a right-tailed test consists of large positive z-values — not large negative ones.