Rejection Region Diagram

See exactly where the rejection region sits for a z or t test, and whether your observed test statistic lands inside it.

Your test

Test statistic

Alternative hypothesis

Significance α

Observed z

Standard normal — rejection region shaded

Critical values

±1.96

Rejection area

p-value

0.0124

Reject H₀. The observed z of 2.5 falls inside the rejection region, and the p-value (0.0124) is below α = 0.05. The two ways of deciding always agree.

The shaded red area is the significance level α: the probability of rejecting H₀ when it is actually true (a Type I error).

How to use this tool

Choose z or t, and whether the alternative hypothesis is two-tailed, right-tailed, or left-tailed.
Set the significance level α and, for a t test, the degrees of freedom.
Enter your observed test statistic. It appears as a black line on the diagram.
Read the critical values, the p-value, and the decision. Change α to watch the shaded area grow and shrink.

Formula Reference

Two-tailed critical values

±c where P(T > c) = α ÷ 2

Right-tailed

c where P(T > c) = α

p-value (two-tailed)

2 × P(T > |t|)

Decision rule

Reject H₀ if the statistic is in the rejection region, or if p < α

ArticleThe Hypothesis Testing Framework ArticleOne-Tailed vs. Two-Tailed Tests Articlet-Distribution vs. Normal Calct-Test Calculator ToolBell Curve Explorer GlossaryGlossary: Critical Value

Frequently Asked Questions

What is a rejection region?+

It is the set of values of the test statistic that would lead you to reject the null hypothesis. Its total probability under the null hypothesis equals the significance level α.

Why does the p-value decision always match the rejection region decision?+

They are two views of the same rule. The observed statistic lies inside the rejection region exactly when the tail area beyond it, the p-value, is smaller than α.

Why is the t distribution wider than the normal one?+

A t statistic uses an estimated standard deviation, which adds extra uncertainty, especially with few degrees of freedom. Its tails are heavier, so its critical values are larger. As degrees of freedom grow, it approaches the normal curve.

What is the shaded area?+

It is α, the probability of a Type I error: rejecting the null hypothesis when it is actually true. In a two-tailed test, α is split equally between the two tails.

How to use this tool

Choose z or t, and whether the alternative hypothesis is two-tailed, right-tailed, or left-tailed.

Set the significance level α and, for a t test, the degrees of freedom.

Enter your observed test statistic. It appears as a black line on the diagram.

Read the critical values, the p-value, and the decision. Change α to watch the shaded area grow and shrink.