Binomial hypothesis testing is the standard A-level method for deciding whether an observed count of successes is consistent with a claimed probability. Whenever a question describes a fixed number of trials, two outcomes (success or failure), a constant probability of success and independent trials, a binomial hypothesis test is the right tool. Typical contexts include manufacturing defect rates, biased coins or dice, double-yolk eggs and survey response rates.
Most marks at A-level are lost on four things: finding the upper critical region, mishandling the discrete boundary values, mixing up the nominal and actual significance level, and writing a conclusion that is either assertive or out of context. This guide walks through how to set up the hypotheses, identify the tail, compute critical regions from the cumulative tables in the Edexcel formulae booklet, and write a valid conclusion — with a worked binomial hypothesis test at the end. If you are testing a sample correlation, see PMCC hypothesis testing; if the question concerns a population mean, see normal distribution hypothesis testing.
A binomial hypothesis test compares a single observed count of successes against the distribution that would be expected under a claimed probability. The null hypothesis fixes the probability of success at a specific value p₀; the alternative hypothesis is that the true value of p is different, higher, or lower. Unlike PMCC hypothesis testing, binomial tests are about counts of success rather than correlation between two variables.
Use a binomial test when all four binomial conditions are explicit or strongly implied:
If the data are paired observations of two quantitative variables, the question is asking for correlation, not proportion – use a PMCC test. If the question is about a sample mean from a normal population, use hypothesis tests on a population mean instead.
The null hypothesis always pins p to a specific value. The alternative hypothesis depends on the direction stated in the question:
| Question wording | Hypotheses |
|---|---|
| “…has the probability changed / is the coin biased / does the defect rate differ?” | H₀: p = p₀, H₁: p ≠ p₀ |
| “…has the probability increased / is the proportion higher?” | H₀: p = p₀, H₁: p > p₀ |
| “…has the probability decreased / is the proportion lower?” | H₀: p = p₀, H₁: p < p₀ |
Always define p in words alongside the symbols, e.g. “Let p denote the probability that a randomly selected egg has a double yolk.”
Reading the question precisely is what determines the tail. In a binomial setting, look for these signals:
For a two-tailed test at the 5% level, split the significance evenly between the tails: each tail should be as close as possible to 2.5% (some questions ask for “less than 2.5%” — read the criterion carefully).
The binomial distribution is discrete, which is what makes the upper tail harder than the lower tail. The Edexcel formula booklet gives cumulative probabilities P(X ≤ x) for selected values of n and p. Use them like this:
A useful classroom habit is to write down P(X ≤ c) for two or three candidate values of c before picking the final boundary. This prevents the common error of using a value one above or one below the correct boundary.
The actual significance level of a binomial test is the sum of the actual tail probabilities for the chosen critical region — not the nominal 5% or 10%. Recent papers often phrase this as: “State the probability of incorrectly rejecting H₀.” The two phrases refer to the same quantity.
For a two-tailed test this is P(X ≤ c₁) + P(X ≥ c₂). For a one-tailed test it is the single tail probability. Always give the value to the precision specified at the front of the paper.
Two video tutorials walk through the full method — carrying out a one-tail and two-tail test, then finding critical regions using the cumulative tables in the formulae booklet.
Students are introduced to hypothesis testing with the binomial distribution. They learn to set up null and alternative hypotheses, calculate probabilities, understand significance levels and draw conclusions within the context of the problem.
Later, they apply this knowledge to real-life problems. They test manufacturing defect rates, evaluate agricultural claims, and assess sales performance.
Students learn how to find the critical region for a binomial distribution hypothesis test. They begin with one-tail tests, using cumulative distribution tables to determine when to reject the null hypothesis.
As their understanding deepens, they move on to two-tail tests, incorporating calculator methods to handle more complex scenarios. Finally, students apply these techniques to real-life situations, such as testing manufacturing quality, evaluating medical treatments, and assessing market research claims.
Tutorials
Finding Critical Regions For One & Two Tail Hypothesis Tests
Meghan rolls an eight-sided dice 120 times and finds that it lands on the number eight 25 times. Use a two-tailed test with a significance level of 5% to determine whether there is sufficient evidence to conclude that the dice is biased.
Worked Solution
Let p denote the probability that a single roll of the dice shows an eight. Under fair behaviour, p = 1/8 = 0.125.
H₀: p = 0.125
H₁: p ≠ 0.125 (two-tailed at the 5% level, so 2.5% in each tail)
Let X denote the number of eights in 120 rolls. Under H₀, X ~ B(120, 0.125), so the expected number of eights is 120 × 0.125 = 15.
The observed value is in the upper tail (25 > 15), so calculate P(X ≥ 25) = 1 − P(X ≤ 24). From a calculator, P(X ≤ 24) ≈ 0.9957, giving P(X ≥ 25) ≈ 0.0043.
The upper-tail significance is 0.025. Since 0.0043 < 0.025, the result falls within the critical region.
Reject H₀. There is sufficient evidence at the 5% level to suggest that the dice is biased in favour of the number eight.
For further practice: a chicken farmer knows that one egg in every 50 has a double yolk. After changing feed, she samples 50 eggs and finds 3 with double yolks. Test at the 10% level whether the chance of a double yolk has changed. (Solution structure: H₀: p = 0.02, H₁: p ≠ 0.02, model X ~ B(50, 0.02), compute P(X ≥ 3) and compare with 0.05.)
Two ready-to-teach lessons cover the procedural and critical-region sides of binomial hypothesis testing in detail, with scaffolded examples, fully worked solutions and differentiated practice.
Mr Mathematics members can download the PowerPoint, differentiated worksheet and student PDF for both lessons.
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