Rejection Region: Definition, Formula and How to Find It

A rejection region (also called the critical region) is the set of test-statistic values that lead you to reject the null hypothesis. If your computed statistic lands inside it, you reject H₀; if it lands outside, you do not. The region is fixed before you look at the data, and its total probability under H₀ equals the significance level α.

Definition in one paragraph

The NIST/SEMATECH e-Handbook of Statistical Methods defines the idea through critical values: they are “cut-off values that define regions where the test statistic is unlikely to lie,” and the null hypothesis is rejected if the test statistic lies within this region, “often referred to as the rejection region(s)” (NIST/SEMATECH e-Handbook, section 7.1.3.1 (Critical values and p values), checked 2026-10-29). The same handbook says the critical region “encompasses those values of the test statistic that lead to a rejection of the null hypothesis,” and that values above, below or both, depending on the direction of the test, define it (NIST/SEMATECH e-Handbook, EDA section on the critical region, checked 2026-10-29). The significance level α sets how large the region is: with α = 0.05, H₀ is rejected 5% of the time when it is in fact true.

Formula: the region for each alternative hypothesis

There is no single formula; the region depends on the sampling distribution of the test statistic and on the direction of the alternative hypothesis H_a. NIST writes the two-sample t-test regions as follows, where T is the test statistic, ν the degrees of freedom and t_p,ν the p-quantile of the t distribution (NIST/SEMATECH e-Handbook, section 1.3.5.3, checked 2026-10-29):

Alternative	Name	Reject H0 when
Ha: μ1 ≠ μ2	Two-tailed	|T| > t1−α/2, ν
Ha: μ1 > μ2	Right-tailed (upper)	T > t1−α, ν
Ha: μ1 < μ2	Left-tailed (lower)	T < tα, ν

The same pattern holds for any test statistic with a known distribution: put the whole α in one tail for a directional alternative, or α/2 in each tail for a two-sided alternative. The boundary of the region is the critical value; see how to find a critical value for lookup tables.

Worked example: a z-test with the arithmetic shown

Suppose H₀: μ = 100, the population standard deviation is known to be σ = 15, a sample of n = 36 gives a mean of 105.5, and α = 0.05.

1. Standard error: σ/√n = 15/√36 = 15/6 = 2.5.
2. Test statistic: z = (105.5 − 100)/2.5 = 5.5/2.5 = 2.20.
3. Right-tailed region (H_a: μ > 100): reject if z > 1.645. Since 2.20 > 1.645, reject H₀.
4. Two-tailed region (H_a: μ ≠ 100): reject if |z| > 1.960. Since 2.20 > 1.960, reject H₀.
5. Left-tailed region (H_a: μ < 100): reject if z < −1.645. Since 2.20 is not below −1.645, do not reject.

The same statistic gives three different decisions or boundaries depending only on the alternative you committed to in advance. Choosing the direction after seeing the data invalidates the stated α. See two-tailed hypothesis testing for a full t-test example and one-tailed vs two-tailed tests for when each is defensible.

Rejection region for z, t, chi-square and F tests

z and t

For z the standard normal quantiles at α = 0.05 are 1.645 (one-tailed) and 1.960 (two-tailed). For t the boundary depends on degrees of freedom: with 35 degrees of freedom, the right-tailed boundary is 1.690 and the two-tailed boundary is 2.030 (computed with the t quantile function). NIST’s two-sample t-test example shows the shift on real data: for the same data and α, the two-tailed critical value is 1.9673 and the upper one-tailed critical value is 1.6495 (NIST/SEMATECH e-Handbook, section 1.3.5.3, checked 2026-10-29).

Chi-square

The chi-square distribution is not symmetric, so the two-tailed region uses two different cut-offs. NIST writes the chi-square test for a variance with T > χ²_{1−α, N−1} for an upper one-tailed alternative, T < χ²_{α, N−1} for a lower one-tailed alternative, and T < χ²_{α/2, N−1} or T > χ²_{1−α/2, N−1} for a two-tailed test (NIST/SEMATECH e-Handbook, section 1.3.5.8, checked 2026-10-29). With 9 degrees of freedom and α = 0.05 the upper one-tailed cut-off is 16.919, the lower is 3.325, and the two-tailed cut-offs are 2.700 and 19.023. Chi-square tests of independence and goodness of fit are almost always run with only the upper-tail region, because large values signal poor fit.

F

The F distribution is also nonnegative and right-skewed. In the common ANOVA F-test the region is the upper tail only: with 4 and 20 degrees of freedom at α = 0.05, reject when F > 2.866.

Interpretation: region, p-value and confidence interval agree

A statistic is in the rejection region exactly when the p-value is at most α. In the example above z = 2.20 gives a one-tailed p of 0.0139 and a two-tailed p of 0.0278; both are below 0.05, matching the decisions above. For a two-sided test the 95% confidence interval for the mean, 105.5 ± 1.96 × 2.5 = (100.6, 110.4), excludes the null value 100, which is the same decision again. To go from z to a p-value, see how to find a p-value from a z-score.

Where the statistic falls	p-value vs α	Decision
Inside the rejection region	p ≤ α	Reject H0
Outside the rejection region	p > α	Fail to reject H0
Exactly on the boundary	p = α	Convention-dependent; state your rule in advance

Assumptions and common mistakes

• The region is only valid if the test’s assumptions hold (for the z-test above: known σ and either a normal population or a large sample).
• Set α and the direction of H_a before seeing the data. Switching from two-tailed to one-tailed afterwards changes the real error rate.
• Falling outside the region means “fail to reject,” not “H₀ is true.”
• Do not use symmetric ± cut-offs for chi-square or F; their two-tailed regions are asymmetric.
• The region is defined on the test statistic, not on the raw sample mean; convert first. See test statistic formula.

How to report it (APA example)

APA style reports the test statistic, degrees of freedom where they exist, and the exact p-value rather than the region itself. For the example above: “A one-sample z-test showed the sample mean (M = 105.5, n = 36) was significantly higher than the hypothesized value of 100, z = 2.20, p = .014 (one-tailed), α = .05.” Report whether the test was one- or two-tailed.

How to calculate it in Excel, R or Python

Excel

=NORM.S.INV(0.95)        // 1.645, right-tailed z cut-off
=NORM.S.INV(0.975)       // 1.960, upper two-tailed z cut-off
=T.INV.2T(0.05, 35)      // 2.030, two-tailed t cut-off
=CHISQ.INV.RT(0.05, 9)   // 16.919, upper chi-square cut-off

R

qnorm(0.95); qnorm(0.975)        # 1.645, 1.960
qt(0.975, df = 35)               # 2.030
qchisq(0.95, df = 9)             # 16.919
qchisq(c(0.025, 0.975), df = 9)  # 2.700, 19.023

Python

from scipy import stats
stats.norm.ppf([0.95, 0.975])        # 1.645, 1.960
stats.t.ppf(0.975, df=35)            # 2.030
stats.chi2.ppf(0.95, df=9)           # 16.919
stats.f.ppf(0.95, dfn=4, dfd=20)     # 2.866

FAQ

Is the rejection region the same as the critical region?

Yes. NIST uses “critical region” and notes the region is often called the rejection region.

How does the rejection region change between one-tailed and two-tailed tests?

A one-tailed test puts all of α in one tail, giving a boundary closer to the center (1.645 for z at 0.05); a two-tailed test splits α across both tails (±1.960).

What is the difference between the rejection region and the p-value?

The region compares the test statistic to a fixed cut-off; the p-value compares a probability to α. They always lead to the same decision.

Does a bigger α make the region larger?

Yes. Raising α moves the cut-off toward the center, so more statistic values fall inside the region, at the cost of more false rejections. See what alpha means in statistics.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, section 7.1.3.1 (Critical values and p values), https://www.itl.nist.gov/div898/handbook/prc/section1/prc131.htm, checked 2026-10-29.
• NIST/SEMATECH e-Handbook, EDA section on the critical region, https://www.itl.nist.gov/div898/handbook/eda/section3/eda35.htm, checked 2026-10-29.
• NIST/SEMATECH e-Handbook, two-sample t-test, https://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm, checked 2026-10-29.
• NIST/SEMATECH e-Handbook, chi-square test for the variance, https://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm, checked 2026-10-29.