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Two-tailed tests

Edexcel IAL Mathematics · WST02 Statistics 2

When the question asks whether a parameter has changed, without saying which way, the test is two-tailed and the significance level is split between the two tails.

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Splitting α

For a two-tailed test at level α, H1 uses ≠ and there are two critical regions, one in each tail. Each tail has probability as close as possible to α/2 without exceeding it. At 5% each tail is at most 0.025; at 10%, at most 0.05. Pearson mark schemes follow this even when a bigger tail is nearer to α/2: for Po(10) at 5% (June 2025) the lower region is X ≤ 3 (0.0103), not X ≤ 4 (0.0293). If a question states its own rule for the tails, follow the question. A common error, noted by examiners, is to use α rather than α/2 in each tail.

Example

H0: p = 0.4, H1: p ≠ 0.4, n = 20, 5% level.

Lower tail: P(X ≤ 3) = 0.0160, P(X ≤ 4) = 0.0510, so X ≤ 3.

Upper tail: P(X ≥ 13) = 1 - 0.9790 = 0.0210, P(X ≥ 12) = 1 - 0.9435 = 0.0565, so X ≥ 13.

Critical region: X ≤ 3 or X ≥ 13. Actual significance level = 0.0160 + 0.0210 = 0.0370.

Using a probability instead

With an observed value x, decide which tail it is in (compare x with the mean np), find the probability of that tail (P(X ≥ x) if x is above the mean, P(X ≤ x) if below), and compare it with α/2.

Worked example

H0: p = 0.4, H1: p ≠ 0.4, n = 20. The observed value is 13. Test at the 5% level.

  1. The mean is 8, so 13 is in the upper tail.
  2. P(X ≥ 13) = 0.0210
  3. 0.0210 < 0.025, so reject H0.

Answer: There is evidence that the proportion has changed.

Key idea: Two-tailed: H1 uses ≠, each tail at most α/2, actual significance level = sum of both tails.

Check yourself

  1. H0: p = 0.4, H1: p ≠ 0.4, n = 20, 5% level. The lower critical region is X ≤ c. Find c.
    Show answer

    3. P(X ≤ 3) = 0.0160 ≤ 0.025 but P(X ≤ 4) = 0.0510 > 0.025.

  2. H0: p = 0.4, H1: p ≠ 0.4, n = 20, 5% level. The upper critical region is X ≥ d. Find d.
    Show answer

    13. P(X ≥ 13) = 0.0210 ≤ 0.025 but P(X ≥ 12) = 0.0565 > 0.025.

  3. For the critical region X ≤ 3 or X ≥ 13 with X ~ B(20, 0.4), find the actual significance level. Give your answer to 4 decimal places.
    Show answer

    0.0370. 0.0160 + 0.0210 = 0.0370.

  4. H0: p = 0.4, H1: p ≠ 0.4, n = 20. The observed value is 4. Find P(X ≤ 4) under H0. Give your answer to 4 decimal places.
    Show answer

    0.0510. P(X ≤ 4) = 0.0510.

  5. In the two-tailed 5% test above, P(X ≤ 4) = 0.0510. What is the decision?
    • Reject H0, because the probability is less than 0.05
    • Reject H0, because 4 is below the mean
    • Do not reject H0, because the probability is more than 0.025
    • Do not reject H0, because the probability is less than 0.5
    Show answer

    Do not reject H0, because the probability is more than 0.025. Compare a single tail with α/2 = 0.025.

  6. A two-tailed test is carried out at the 10% level. What is the largest probability allowed in each tail?
    • 0.05
    • 0.1
    • 0.025
    • 0.2
    Show answer

    0.05. α/2 = 0.05.

  7. In a two-tailed test, the observed value is above the mean. Which probability should be compared with α/2?
    • P(X ≤ observed value)
    • P(X = observed value)
    • P(X ≥ observed value)
    • P(X ≥ mean)
    Show answer

    P(X ≥ observed value). Use the tail that contains the observed value: values at least as extreme.

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