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Critical regions and actual significance levels

Edexcel IAL Mathematics · WST02 Statistics 2

Instead of finding a probability for one observed value, you can work out in advance every value that would lead to rejecting H0. This set is the critical region.

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Definitions

The critical region is the set of values of the test statistic that lead to rejecting H0.
The critical value is the first value inside the critical region.
The acceptance region is everything else.
The actual significance level is the probability of the test statistic falling in the critical region when H0 is true. Because the distributions are discrete, it is usually less than the stated level.

Finding a one-tailed critical region

Upper tail, X ~ B(20, 0.3) under H0, 5% level: find the smallest c with P(X ≥ c) ≤ 0.05.

P(X ≥ 9) = 1 - P(X ≤ 8) = 1 - 0.8867 = 0.1133 (too big)

P(X ≥ 10) = 1 - P(X ≤ 9) = 1 - 0.9520 = 0.0480

Critical region X ≥ 10; actual significance level 0.0480 (4.80%).

Lower tail: find the largest c with P(X ≤ c) ≤ 0.05.

Writing it down

Show the probability statements that lead to your choice (examiners withhold marks for a region with no supporting probabilities). Then write the region as an inequality or set, such as X ≥ 10 or {10, 11, ..., 20}; a probability statement is not a critical region. If the question asks for the critical region, do not give the acceptance region.

Worked example

Under H0, X ~ Po(8). Find the critical region for a test of H0: λ = 8 against H1: λ < 8 at the 5% level, and the actual significance level.

P(X ≤ 3) = 0.0424 (≤ 0.05)
P(X ≤ 4) = 0.0996 (> 0.05)

Answer: Critical region X ≤ 3; actual significance level 0.0424

Key idea: Critical region: values that reject H0, chosen so the tail probability is as large as possible without exceeding α. Actual significance level = P(in critical region | H0).

Check yourself

Under H0, X ~ B(20, 0.3). Find the critical value c for the critical region X ≥ c, testing H1: p > 0.3 at the 5% level.
Show answer
10. P(X ≥ 9) = 0.1133 > 0.05 and P(X ≥ 10) = 0.0480 ≤ 0.05, so c = 10.
For the critical region X ≥ 10 with X ~ B(20, 0.3), find the actual significance level. Give your answer to 4 decimal places.
Show answer
0.0480. P(X ≥ 10) = 1 - 0.9520 = 0.0480.
Under H0, X ~ Po(8). Find the largest c for the critical region X ≤ c, testing H1: λ < 8 at the 5% level.
Show answer
3. P(X ≤ 3) = 0.0424 ≤ 0.05 but P(X ≤ 4) = 0.0996 > 0.05, so c = 3.
For the critical region X ≤ 3 with X ~ Po(8), find the actual significance level. Give your answer to 4 decimal places.
Show answer
0.0424. P(X ≤ 3) = 0.0424.
Under H0, X ~ B(25, 0.4). Find the largest c for the critical region X ≤ c, testing H1: p < 0.4 at the 5% level.
Show answer
5. P(X ≤ 5) = 0.0294 ≤ 0.05; P(X ≤ 6) = 0.0736 > 0.05.
A test has critical region X ≥ 10. The observed value is x = 9. What is the conclusion?
- Reject H0, because 9 is close to 10
- Do not reject H0, because 9 is not in the critical region
- Reject H0, because 9 is greater than the mean
- The test cannot be completed
Show answer
Do not reject H0, because 9 is not in the critical region. Only values in the critical region lead to rejection.
Which is a correctly written critical region?
- P(X ≥ 10) = 0.048
- X ≤ 9
- p > 0.3
- X ≥ 10
Show answer
X ≥ 10. A critical region is a set of values of the test statistic.

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Last updated 8 November 2026. Spotted a mistake? Tell us.