Write a Rejection Region in Interval Notation

A researcher runs a two-tailed Z-test at α=0.05 and gets a test statistic of 2.01. She needs to know if that value falls inside the rejection region. The answer is yes, but only if the rejection region is written as two intervals: (−∞, −1.96] ∪ [1.96, ∞). Writing it as a single interval or with the wrong cutoff will produce a wrong decision. To write a rejection region in interval notation, you express the set of test-statistic values that lead to rejecting the null hypothesis as an interval or a union of intervals.

The rejection region is the set of test-statistic values for which the null hypothesis is rejected. For a one-tailed right test, that set is (critical value, ∞). For a one-tailed left test, it is (−∞, critical value). For a two-tailed test with a symmetric distribution like Z or t, the rejection region is (−∞, −critical value] ∪ [critical value, ∞). The critical value is the quantile from the sampling distribution at the appropriate alpha level, and the notation is the natural final step after you have found those cutoffs.

Work one concrete example from start to interval notation. A two-tailed Z-test at α=0.05 requires the critical value at the 0.975 quantile of the standard normal distribution. That value is Φ⁻¹(0.975) = 1.96. The rejection region is the set of test-statistic values that are at least 1.96 standard deviations from zero in either direction.

In interval notation: (−∞, −1.96] ∪ [1.96, ∞). A test statistic of 2.01 falls in the right interval, so the null is rejected. A test statistic of 1.50 falls between the intervals, so the null is not rejected. The union symbol ∪ is essential: without it, the region reads as a single contiguous interval, which does not exist.

For a one-tailed right Z-test at the same α=0.05, the critical value is Φ⁻¹(0.95) = 1.645. The rejection region is (1.645, ∞). A test statistic of 1.65 falls in that region, but a test statistic of 1.64 does not. The one-tailed region uses a smaller critical value because all of alpha is in one tail.

To compute the critical value and then write the interval, you need the inverse of the cumulative distribution function for your test statistic. For a one-tailed right test, the critical value is F⁻¹(1−α), where F is the CDF of the distribution. For a two-tailed symmetric test, it is F⁻¹(1−α/2).

Z-Test With Known Sigma

When the population standard deviation is known, use the Z-distribution. At α=0.05 two-tailed, the critical value is 1.96. The rejection region is (−∞, −1.96] ∪ [1.96, ∞). This is the most common interval notation in introductory statistics.

T-Test With Estimated Sigma

When sigma is estimated from the sample standard deviation, use the t-distribution with degrees of freedom n−1. For a one-tailed right t-test with df=10 and α=0.05, the critical value is t_{0.95,10} = 1.812. The rejection region is (1.812, ∞). The critical value is larger than the Z critical of 1.645 because the t-distribution has heavier tails.

If you need a critical value for a non-standard alpha level or for a t-test with degrees of freedom not in your printed table, use an interval notation calculator. It will return the exact quantile, which you then write into the interval format.

Chi-Square and F Tests

For asymmetric distributions like chi-square and F, a two-tailed rejection region is not symmetric. It must be written as two separate intervals: [χ²_{α/2, df}, χ²_{1−α/2, df}] for chi-square, or [F_{α/2, df1, df2}, F_{1−α/2, df1, df2}] for F. The left critical value is near zero, not the negative of the right critical value. Using a symmetric notation for these distributions is a common failure case.

The notation is straightforward once you understand what each symbol represents. An interval like (a, b) includes all values between a and b but not the endpoints. A bracket like [a, b] includes the endpoints. The union symbol ∪ connects disjoint intervals.

For a one-tailed left test, the rejection region is (−∞, critical value). The critical value is negative for distributions centered on zero, like Z and t. For a one-tailed right test, the region is (critical value, ∞). The critical value is positive.

For a two-tailed test with a symmetric distribution, the rejection region is (−∞, −critical value] ∪ [critical value, ∞). The critical value is positive. The negative of the critical value marks the left boundary. For asymmetric distributions, you compute two separate critical values and write them as intervals with brackets.

The most common misinterpretation: writing a two-tailed rejection region as a single interval like (−∞, ∞) or (1.96, ∞). The first is the entire real line, which would reject the null for every sample. The second is a one-tailed region, which would only detect deviations in one direction and would inflate the Type I error rate to α instead of α/2 per tail.

Test Type	Distribution	Alpha	Critical Value	Rejection Region
One-tailed right	Z	0.05	1.645	(1.645, ∞)
One-tailed left	Z	0.05	−1.645	(−∞, −1.645]
Two-tailed	Z	0.05	1.96	(−∞, −1.96] ∪ [1.96, ∞)
One-tailed right	t (df=10)	0.05	1.812	(1.812, ∞)
Two-tailed	t (df=10)	0.05	2.228	(−∞, −2.228] ∪ [2.228, ∞)

The most common failure is using a one-tailed rejection region for a two-tailed test. A researcher tests for a difference in either direction but writes the rejection region as (1.645, ∞) instead of (−∞, −1.96] ∪ [1.96, ∞). The result: any test statistic above 1.645 rejects the null, but the actual two-tailed critical value at α=0.05 is 1.96. The Type I error rate inflates to 0.10 per tail, double the intended α.

Another failure case: using Z when sigma is unknown. A student with n=35 uses the Z critical value of 1.96 but should use a t critical value because the population standard deviation is estimated from the sample. The t critical value for df=34 at α=0.05 two-tailed is 2.032. The rejection region (−∞, −1.96] ∪ [1.96, ∞) is too narrow, leading to more Type I errors than intended.

A limitation of interval notation: it assumes the test statistic is continuous. For discrete distributions, the rejection region may need to be defined by integer thresholds. Only continuous distributions are covered.

The single thing that most often goes wrong: using a left-tail critical value from a chi-square or F table instead of the right-tail value. A chi-square goodness-of-fit test at α=0.05 with df=5 requires a right-tail critical value of 11.07. Using the left-tail value of 0.831 produces a rejection region that is almost the entire real line, rejecting the null for nearly every sample.

High-school AP Statistics students who need to write rejection regions for one- and two-tailed tests using Z and t distributions are suited for this content. It also suits college intro statistics students who must verify the output of a statistical package by writing the region in interval format.

Graduate students in applied fields like psychology and public health who need exact critical values for non-standard alpha levels will find the procedure here. Analysts checking spreadsheet functions like T.INV or NORM.S.INV will also benefit.

Skip this analysis if you need to compute a p-value directly instead of comparing a test statistic to a threshold. Use a p-value calculator instead. Skip if you need critical values for non-parametric tests like Mann-Whitney U or Wilcoxon signed-rank; those distributions are not covered here.