The rejection region is the set of outcomes in a hypothesis test that lead you to reject the null hypothesis. Think of it as a boundary line drawn on a distribution curve before you collect data. If your test result lands in that zone, you conclude there’s enough evidence against the null hypothesis. If it doesn’t, you fail to reject it. The size of this zone is determined by your chosen significance level, commonly set at 0.05 (5%).
How the Rejection Region Works
In hypothesis testing, you start with a null hypothesis, which is typically a statement that nothing unusual is happening (no difference between groups, no effect of a treatment). You then collect data and calculate a test statistic, which is a single number summarizing how far your results deviate from what the null hypothesis predicts.
The rejection region is the range of test statistic values that would be so extreme, so unlikely under the null hypothesis, that you’d conclude the null hypothesis is probably wrong. The boundary of this region is called the critical value. If your test statistic is more extreme than the critical value, it falls in the rejection region and you reject the null hypothesis. If it’s less extreme, you don’t.
Here’s the key idea: you define the rejection region before looking at your data. This keeps the decision rule objective. You’re essentially saying, “If the data are this surprising or more, I’ll reject the null hypothesis.”
Alpha Sets the Size of the Region
The significance level, called alpha, controls how large the rejection region is. If you set alpha at 0.05, the rejection region covers exactly 5% of the area under the distribution curve. That means there’s a 5% chance your test statistic would land in the rejection region purely by random chance, even if the null hypothesis were true.
A stricter alpha, like 0.01, shrinks the rejection region to just 1% of the curve. This makes it harder to reject the null hypothesis, but it also means you’re less likely to make a false alarm. The tradeoff is straightforward: a smaller rejection region protects against false positives but makes it easier to miss real effects.
The probability of incorrectly rejecting a true null hypothesis is called a Type I error, and alpha is exactly that probability. So when you choose alpha = 0.05, you’re accepting a 5% maximum chance of concluding there’s an effect when there actually isn’t one.
One-Tailed vs. Two-Tailed Tests
Where the rejection region sits on the distribution depends on what kind of alternative hypothesis you’re testing. This is where directionality matters.
- Right-tailed test: Your alternative hypothesis predicts a value greater than the null. The entire rejection region sits in the right tail of the distribution. At alpha = 0.05, the critical value is 1.64 (for a z-test), and any test statistic above that falls in the rejection region.
- Left-tailed test: Your alternative hypothesis predicts a value less than the null. The entire rejection region sits in the left tail. The critical value is -1.64 at alpha = 0.05.
- Two-tailed test: Your alternative hypothesis simply predicts a difference in either direction. The rejection region is split between both tails, with half the alpha in each. At alpha = 0.05, that’s 2.5% in each tail, giving critical values of ±1.96. Your test statistic needs to exceed 1.96 or fall below -1.96 to land in the rejection region.
Two-tailed tests are more common because researchers often can’t predict the direction of an effect in advance. The cost is that the critical values are farther from the center, making it slightly harder to reject the null hypothesis compared to a one-tailed test at the same alpha.
Common Critical Values to Know
For a standard normal (z) distribution, the critical values you’ll encounter most often in a two-tailed test are ±1.96 at the 0.05 significance level and ±2.575 at the 0.01 level. These numbers come directly from the properties of the normal curve: 95% of the area falls within ±1.96 standard deviations of the mean, leaving 5% in the tails.
When you’re working with smaller samples and using a t-distribution instead of a normal distribution, the critical values shift outward. The t-distribution has heavier tails than the normal distribution, meaning extreme values are more likely. With 20 degrees of freedom and alpha = 0.05 in a two-tailed test, the critical value is about ±2.08 rather than ±1.96. As your sample size grows, the t-distribution approaches the normal distribution. A common rule of thumb is that with 30 or more observations, the two are close enough that the z critical values work as a reasonable approximation.
Rejection Region vs. P-Value
There are two ways to reach the same decision in hypothesis testing, and they always agree. The rejection region approach sets a critical value in advance and checks whether the test statistic crosses it. The p-value approach calculates the exact probability of getting a result as extreme as (or more extreme than) what you observed, then checks whether that probability is below alpha.
Consider a two-tailed z-test at alpha = 0.05. The rejection region is any z-score beyond ±1.96. If your test statistic is z = 2.5, it falls in the rejection region, so you reject the null hypothesis. Equivalently, the p-value for z = 2.5 is 0.012. Since 0.012 is less than 0.05, you also reject. Both methods are telling you the same thing: the observed data are too unlikely under the null hypothesis to be explained by chance alone.
Now imagine your test statistic is z = 2.1 in the same scenario but with a one-tailed rejection region starting at 1.96. It falls inside the rejection region, and the p-value would confirm that. But if z = 2.1 in a setup where the critical value is 2.575 (alpha = 0.01), it falls outside, and the p-value would be larger than 0.01. The decision flips depending on how strict your threshold is.
The p-value gives you more information because it tells you exactly how extreme the result is, not just whether it crossed a line. The rejection region approach is more rigid but makes the decision binary and straightforward.
A Practical Example
Suppose you want to test whether a new teaching method changes exam scores compared to the traditional method, where the average score is 75. You have no reason to predict whether scores will go up or down, so you choose a two-tailed test with alpha = 0.05. Your rejection region is z less than -1.96 or z greater than 1.96.
You collect data from 36 students, calculate the test statistic, and get z = 2.3. That value is greater than 1.96, placing it squarely in the rejection region. You reject the null hypothesis and conclude there’s statistically significant evidence that the new teaching method produces different exam scores. Had you gotten z = 1.5, it would fall outside the rejection region, and you’d fail to reject the null hypothesis. That doesn’t mean the null hypothesis is true. It means your data weren’t extreme enough to rule it out at the 5% level.
Choosing a different alpha would have changed the outcome. At alpha = 0.01, the critical values widen to ±2.575, and your z = 2.3 would no longer fall in the rejection region. The same data, the same test statistic, but a different conclusion, all because you moved the boundary. This is why the choice of alpha matters and why it should be set before collecting data, not adjusted after the fact to get the result you want.