Rejection Region vs P-Value

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.

A p-value is a conditional probability computed while the null hypothesis is taken as true. Written out, it is $P(\text{a result at least as extreme as the observed one} \mid H_0 \text{ is true})$, where extreme means in whichever direction the alternative hypothesis points.

Suppose you flip a coin 100 times, get 60 heads, and test $H_0: p = 0.5$ against $H_a: p \ne 0.5$. Under the null, $\hat{p}$ (p hat) is centered at 0.5 with standard deviation 0.05, so the observed $\hat{p} = 0.60$ sits $z = \frac{0.60 - 0.50}{0.05} = 2.00$ standard deviations out. Under the normal model the z-test uses, the two-sided p-value is 0.0455. Said aloud: if the coin really were fair, that model puts about 4.6 percent of samples of 100 flips at least 0.10 away from 0.5 in one direction or the other. Against the one-sided $H_a: p > 0.5$ the same model gives 0.0228, half as much, because only the upper tail counts.

Two readings of that 0.0455 are wrong and both are everywhere. It is not the probability that the coin is fair, and it is not the probability that the result happened by chance. Both of those are claims about the hypothesis, while the p-value is a claim about the data that takes the hypothesis as given. The conditioning runs one way only, and reversing it is the single most repeated error in applied statistics.

A large p-value is not evidence for $H_0$ either. It means the data would not be surprising if $H_0$ were true, which is a much weaker statement than $H_0$ being true. Small samples have little power to detect anything, so a large p-value from one may mean only that the test could not have found an effect of ordinary size, which is why a p-value of 0.4 from 12 observations tells you very little.

The decision comes from comparing the p-value against a significance level $\alpha$ (alpha) fixed before the data are seen: reject $H_0$ when the p-value is at or below $\alpha$, and fail to reject it otherwise. The p-value itself is a measure of evidence on a continuous scale, so 0.049 and 0.051 carry nearly identical information despite falling on opposite sides of the usual cutoff.