Rejection region

A rejection region is fixed by three things, all settled before the data arrive: the significance level $\alpha$ (alpha), the direction of $H_a$, and the reference curve the statistic will be read against. Find the critical value that cuts off area $\alpha$ in the relevant tail or tails, and the region is everything at or beyond it. It is a set of test statistic values, not a set of parameter values.

For a two-sided $z$ test at $\alpha = 0.05$ the cutoff is $z^* = 1.960$, so the region is $|z| \ge 1.960$ with 0.025 in each tail. Make the alternative one-sided and the whole 5% moves to one side, dropping the cutoff to 1.645: the region is $z \ge 1.645$, or $z \le -1.645$ for a left-tailed $H_a$. Change the curve to a $t$ with 24 degrees of freedom and the two-sided cutoff rises to 2.064, because a $t$ curve carries heavier tails than the normal.

Which is why "anything past 1.96 is significant" costs points. A statistic of 2.00 rejects on the standard normal, two-sided, at $\alpha = 0.05$. The same 2.00 fails to reject on a $t$ curve with 24 degrees of freedom, where the cutoff is 2.064 and the two-sided p-value is 0.0569. Memorising the number instead of the recipe is how correct arithmetic ends up with the wrong verdict attached.

Not every region has two sides available. A chi-square test for homogeneity or independence squares its differences, so direction is gone and the region is always the upper tail: with one degree of freedom at $\alpha = 0.05$ it is $\chi^2 \ge 3.841$. And because the region lives on the statistic, it never doubles as a range of plausible values for $p$ or $\mu$ (mu). That job belongs to a confidence interval.

The region and the p-value never disagree, since a statistic lands in the region exactly when its p-value is at or below $\alpha$. The Fall 2026 topic list has a topic titled p-Values (3.6) and none titled rejection region or critical region, so the p-value comparison is the form you meet by name.