Pearson Edexcel · A-Level · Mathematics
Course version: 9MA0 · Version from Mar 2026
Conduct a statistical hypothesis test for the proportion in the binomial…
Full specification wording
Statistical hypothesis testing: 5.2 Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context. Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis.
To test a claim about a proportion p, take a random sample of size n and count successes X. Under H₀, X ~ B(n, p₀), where p₀ is the claimed proportion.
Checked against the Pearson Edexcel 9MA0 specification (opens in a new tab) on .
- Point 2 of 3
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
To test a claim about a proportion p, take a random sample of size n and count successes X. Under H₀, X ~ B(n, p₀), where p₀ is the claimed proportion. Choose H₁ as p > p₀, p < p₀ or p ≠ p₀ to match the claim, and fix the significance level α. Find the critical region using cumulative binomial probabilities, or find the p-value as the probability of a result at least as extreme as the observed X. Compare the observed statistic with the critical region, or the p-value with α, then reject or do not reject H₀. The sample gives evidence about the unknown population proportion, so conclusions are inferences, not certainties. The significance level is the probability of incorrectly rejecting H₀ when it is true.
What earns the marks
- Sets up H₀ and H₁ as statements about the population proportion p, matching the direction of the claim.
- States the binomial model X ~ B(n, p₀) under H₀ and identifies the observed test statistic from the sample.
- Finds the critical region or p-value using cumulative binomial probabilities, with the correct tail or tails for the chosen test.
- Compares the statistic with the critical region, or the p-value with α, and states the decision clearly.
- Writes a conclusion in context that links the decision back to the population proportion and acknowledges sampling uncertainty.
Common mistakes and how to put them right
Using the sample proportion in the binomial model under H₀.
Put it right by using the claimed proportion p₀ in B(n, p₀).
Forgetting to double the tail probability in a 2-tail test.
Put it right by splitting α between both tails.
Concluding that H₀ is proved true when it is not rejected.
Put it right by saying there is insufficient evidence against H₀.
Exam tips
- Write down n, p₀, the observed value and α before calculating.
- Show the cumulative probability calculation or calculator input that gives the critical region or p-value.
- End with a sentence in context that answers the original question and mentions the evidence.
Try the lesson
The start of the Statistical hypothesis testing: 5.2 Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context. Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis. lesson, made from checked teaching notes for this specification point.
Have a go first
A test at the 5% significance level uses X ~ B(20, p) with null hypothesis p = 0.3 and alternative hypothesis p > 0.3. The critical region is X ≥ 10. Find the actual probability of incorrectly rejecting the null hypothesis.
Listen
To test a claim about a proportion p, take a random sample of size n and count successes X. Under H₀, X ~ B(n, p₀), where p₀ is the claimed proportion. Choose H₁ as p > p₀, p < p₀ or p ≠ p₀ to match the claim, and fix the significance level α. Find the critical region using cumulative binomial probabilities, or find the p-value as the probability of a result at least as extreme as the observed X. Compare the observed statistic with the critical region, or the p-value with α, then reject or do not reject H₀. The sample gives evidence about the unknown population proportion, so conclusions are inferences, not certainties. The significance level is the probability of incorrectly rejecting H₀ when it is true. A strong answer earns marks for these points. Sets up H₀ and H₁ as statements about the population proportion p, matching the direction of the claim. States the binomial model X ~ B(n, p₀) under H₀ and identifies the observed test statistic from the sample. Finds the critical region or p-value using cumulative binomial probabilities, with the correct tail or tails for the chosen test. Compares the statistic with the critical region, or the p-value with α, and states the decision clearly. A common slip: Using the sample proportion in the binomial model under H₀. Instead, using the claimed proportion p₀ in B(n, p₀). Exam tip: Write down n, p₀, the observed value and α before calculating.
Key words
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Binomial hypothesis tests for a proportion
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Critical regions and p-values
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Inference from sample to population
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Significance level as a probability of incorrect rejection
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See every definition
- Binomial hypothesis tests for a proportion
- A test of a claim about a proportion p that takes a random sample of size n and counts successes X, with X ~ B(n, p₀) under H₀.
- Critical regions and p-values
- The critical region is found using cumulative binomial probabilities, and the p-value is the probability of a result at least as extreme as the observed X.
- Inference from sample to population
- The sample gives evidence about the unknown population proportion, so conclusions are inferences, not certainties.
- Significance level as a probability of incorrect rejection
- The probability of incorrectly rejecting H₀ when it is true.
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