The critical region is the set of test statistic values that leads you to reject the null hypothesis in a hypothesis test. In Intro to Statistics, it sits in the tail or tails of the sampling distribution.
The critical region is the part of the sampling distribution where your test statistic is considered extreme enough that you reject the null hypothesis. In Intro to Statistics, you use it in hypothesis tests as the cutoff area that separates results that are ordinary under the null from results that look unlikely if the null were true.
Think of it as a decision zone. If your test statistic lands inside that zone, your sample result is so far out in the tail that it gives evidence against H0. If it lands outside the critical region, you do not reject H0, which does not mean the null is proven true. It just means the evidence is not strong enough to cross the cutoff.
The size and location of the critical region depend on the significance level, α, and on the alternative hypothesis. A smaller α makes the rejection region harder to reach because you are demanding stronger evidence before rejecting the null. A one-tailed test puts the region in one tail only, while a two-tailed test splits the rejection area across both tails.
A common way to build the critical region is to start with α and then find the critical value or values that leave exactly that much area in the tail(s). For example, if α = 0.05 in a right-tailed test, the critical region is the upper 5% of the distribution. Any test statistic larger than the cutoff falls there and leads to rejection.
This is why the critical region is tied to the logic of hypothesis testing, not just to a formula. You are not asking whether the sample result is unusual in a casual sense. You are asking whether it is unusual enough, under a specific null model, to fall into the rejection region you set before looking at the data.
The critical region is the step that turns a hypothesis test from a calculation into a decision. In Intro to Statistics, you are often asked to compare a test statistic to a critical value, mark the rejection area on a graph, or explain why a result does or does not justify rejecting H0.
It also connects the ideas of significance level and Type I error. When you choose α, you are choosing how much risk of a false alarm you are willing to accept, and that choice determines how wide the critical region is. A stricter α means a smaller chance of rejecting a true null, but it also makes rejection harder.
You will see this term whenever you interpret p-values, because p-values and critical regions tell the same story in two different ways. The critical region is the cutoff approach: if the statistic enters that region, reject. The p-value approach is the probability approach: if the p-value is smaller than α, reject.
It also helps you keep the logic of one-tailed and two-tailed tests straight. Many mistakes come from shading the wrong tail or using the wrong direction for the alternative hypothesis. Knowing where the critical region belongs keeps your hypothesis test setup clean from the start.
Keep studying Intro to Statistics Unit 9
Visual cheatsheet
view galleryNull Hypothesis
The critical region is defined relative to the null hypothesis, because the whole sampling distribution is built assuming H0 is true. If your test statistic falls in the rejection area, you are saying the data are too extreme to fit the null model well. So the critical region only makes sense once you know what claim you are testing against.
Alternative Hypothesis
The alternative hypothesis tells you which tail or tails to use for the critical region. A right-tailed alternative puts the rejection area on the high end, a left-tailed alternative puts it on the low end, and a two-sided alternative splits α across both sides. If you match the wrong alternative, you shade the wrong region.
Test Statistic
Your test statistic is the value you compare to the critical region. It might be a z-score, t-score, or another standardized value, depending on the test. The whole decision depends on whether that statistic lands inside the rejection area or stays in the nonrejection area.
level of significance
The level of significance, α, sets the size of the critical region before you run the test. A smaller α means the rejection area shrinks, so it takes more extreme evidence to reject H0. In practice, α is the threshold that controls how strict your hypothesis test is.
A problem set question may give you a significance level, a tail direction, and a test statistic, then ask whether to reject the null. Your job is to identify the critical region, compare the statistic to the cutoff, and state the decision clearly. If the test is one-tailed, you only shade one tail; if it is two-tailed, you split α across both tails.
You may also be asked to explain the decision in words, not just mark a graph. A strong response says whether the statistic falls in the rejection region and ties that back to the null hypothesis. If a p-value is given instead, you can still connect it to the critical region by checking whether p-value < α. Many quiz mistakes come from using the wrong tail or confusing the rejection region with the full sampling distribution.
The critical region is a cutoff area in the tail(s) of the distribution, while the p-value is a probability computed from the observed statistic. The critical region tells you where to reject, and the p-value tells you how much evidence you have against H0. They lead to the same decision when you compare the p-value to α.
The critical region is the set of test statistic values that makes you reject the null hypothesis.
Its location depends on the alternative hypothesis, so the rejection area may be in one tail or split across two tails.
The size of the critical region is controlled by the significance level, α.
A statistic inside the critical region is considered extreme enough to count as evidence against H0.
The critical region gives you a visual decision rule for hypothesis testing, especially when you are drawing shaded sampling distributions.
The critical region is the rejection area in a hypothesis test. If your test statistic lands there, you reject the null hypothesis because the sample result is too extreme to fit the null model well. It is usually placed in the tail or tails of the sampling distribution.
Start with the significance level, α, and the type of alternative hypothesis. Then find the cutoff value or values that leave α area in the tail or tails. For a one-tailed test, all of α goes into one tail, and for a two-tailed test, it is split between both tails.
No. The critical region is a range of test statistic values, while the p-value is a probability based on the observed result. They are connected because both are used to make the reject or do-not-reject decision, but they are not the same thing.
You do not reject the null hypothesis. That does not prove the null is true, it only means your sample did not produce evidence strong enough to cross the rejection cutoff. In class problems, that usually shows up as a nonrejection decision with a short interpretation statement.