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Hypothesis tests for a binomial proportion

Edexcel IAL Mathematics · WST02 Statistics 2

Here you put the ideas together: a full one-tailed test of the binomial parameter p, written the way mark schemes expect.

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The standard layout

1. Define p in context and write H0: p = p0 and H1: p > p0 (or p < p0).

2. State the distribution of the test statistic under H0: X ~ B(n, p0).

3. Either find P(X ≥ x) (or P(X ≤ x)) for the observed x and compare with α, or find the critical region and see whether x is in it.

4. Conclude: a statement (reject H0 / do not reject H0) and a sentence in context using the wording of the question.

Worked layout

A shop says 30% of customers use a loyalty card. After a campaign, 9 of 20 customers do. Test at 5% whether the proportion has increased.

H0: p = 0.3, H1: p > 0.3. Under H0, X ~ B(20, 0.3).

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

0.1133 > 0.05, so do not reject H0. There is insufficient evidence to suggest that the proportion of customers using a loyalty card has increased.

Lower tails and p > 0.5

For H1: p < p0, find P(X ≤ x). If p0 is more than 0.5, count failures: for seeds with claimed germination rate 0.85, 13 germinating out of 20 means 7 failures, and P(X ≤ 13) = P(Y ≥ 7) with Y ~ B(20, 0.15).

Questions may also ask for the smallest sample size for which a result could be significant: for H1: p < 0.25 at 5%, X = 0 is significant when 0.75n ≤ 0.05, so n ≥ 11.

Worked example

A claim says 85% of seeds germinate. In a sample of 20, 13 germinate. Test at 5% whether the claim overstates the germination rate.

  1. H0: p = 0.85, H1: p < 0.85. Under H0, X ~ B(20, 0.85).
  2. Y = 20 - X ~ B(20, 0.15) counts failures.
  3. P(X ≤ 13) = P(Y ≥ 7) = 1 - P(Y ≤ 6) = 1 - 0.9781 = 0.0219
  4. 0.0219 < 0.05, so reject H0.

Answer: There is evidence to suggest that the proportion of seeds that germinate is less than 0.85.

Key idea: Define p, write H0 and H1, state X ~ B(n, p0), compare the tail probability (or critical region) with α, conclude in context.

Check yourself

  1. H0: p = 0.3, H1: p > 0.3. In a sample of 20, x = 9. Find P(X ≥ 9) under H0. Give your answer to 4 decimal places.
    Show answer

    0.1133. 1 - P(X ≤ 8) = 1 - 0.8867 = 0.1133.

  2. H0: p = 0.3, H1: p > 0.3 at the 5% level. P(X ≥ 9) = 0.1133. What is the conclusion?
    • Reject H0: evidence that the proportion has increased
    • Do not reject H0: insufficient evidence that the proportion has increased
    • Accept H1: the proportion has increased
    • Reject H0: the proportion has not changed
    Show answer

    Do not reject H0: insufficient evidence that the proportion has increased. 0.1133 > 0.05, so the result is not significant.

  3. H0: p = 0.2, H1: p < 0.2. In a sample of 25, x = 1. Find P(X ≤ 1) under H0. Give your answer to 4 decimal places.
    Show answer

    0.0274. P(X ≤ 1) = 0.0274.

  4. H0: p = 0.85, H1: p < 0.85. In a sample of 20, 13 successes are seen. Find P(X ≤ 13) under H0. Give your answer to 4 decimal places.
    Show answer

    0.0219. Failures Y ~ B(20, 0.15): P(Y ≥ 7) = 1 - 0.9781 = 0.0219.

  5. A test of H0: p = 0.25 against H1: p < 0.25 uses the 5% level. Find the smallest sample size n for which X = 0 would lead to rejecting H0.
    Show answer

    11. Need 0.75n ≤ 0.05: 0.7510 = 0.0563, 0.7511 = 0.0422, so n = 11.

  6. In a test about a loyalty card, which definition of p is best?
    • p is the proportion of all customers who use a loyalty card
    • p is the number of customers in the sample who use a card
    • p is the probability that the test is significant
    • p is 0.3
    Show answer

    p is the proportion of all customers who use a loyalty card. p is the population proportion, in context.

  7. Which contextual conclusion is worded as mark schemes expect when H0 is not rejected?
    • The proportion of customers using a card is 0.3
    • There is evidence that the proportion has not changed
    • There is insufficient evidence to suggest that the proportion of customers using a card has increased
    • H0 is proved true
    Show answer

    There is insufficient evidence to suggest that the proportion of customers using a card has increased. Mark schemes want 'insufficient evidence' with the context, not certainty.

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