Hypothesis testing for A-level Maths

Hypothesis testing in A-level Maths is a way of using a sample to decide whether a claim about a population is believable. You assume the claim is true (the null hypothesis), work out how likely your result would be if it were, and reject the claim if that probability is below the significance level. At A-level you test a binomial proportion, the mean of a normal distribution, and a correlation coefficient.

The language of hypothesis testing

Pearson Edexcel's A-level specification lists the terms you need to understand and use:

Edexcel's guidance says hypotheses should be written in terms of the population parameter: p for a binomial proportion, μ for a normal mean, ρ for a population correlation coefficient. The sample is being used to make an inference about the population.

How to do a hypothesis test in 5 steps

1. Define the variable and its distribution under H₀, such as X ~ B(25, 0.1).
2. Write H₀ and H₁ using the parameter, and decide whether the test is 1-tail or 2-tail.
3. Find the probability of the observed result or anything further from what H₀ predicts (the p-value), or find the critical region.
4. Compare with the significance level. For a 2-tail test, compare each tail with half the significance level.
5. Conclude in context: say whether there is sufficient evidence to reject H₀, and what that means for the claim.

Binomial hypothesis test

Example 1: a one-tail test

A ferry company says 10% of its crossings are delayed. A regular passenger thinks the true figure is higher. In a random sample of 25 crossings, 6 are delayed. Test the claim at the 5% significance level.

1. Let X be the number of delayed crossings in the sample. Under H₀, X ~ B(25, 0.1).
2. H₀: p = 0.1 and H₁: p > 0.1. This is a 1-tail test.
3. The result is 6, so find the probability of 6 or more: P(X ≥ 6) = 1 − P(X ≤ 5) = 1 − 0.9666 = 0.0334.
4. 0.0334 is below 0.05, so H₀ is rejected.
5. There is sufficient evidence at the 5% level that more than 10% of crossings are delayed.

The chart shows the distribution assumed under H₀, with the probability of each number of delays as a percentage. The pink stems, 6 and above, are the critical region: together they have a probability of only 3.34%.

Critical regions

Instead of a p-value you can find the critical region first. For Example 1, look for the smallest c with P(X ≥ c) ≤ 0.05:

So the critical region is X ≥ 6, and the critical value is 6. Because the distribution is discrete, the chance of a result falling in the critical region, the actual significance level, is 3.34%, not exactly 5%.

Example 2: a two-tail test

A library says 65% of its loans are e-books. A librarian thinks the proportion has changed. She will take a random sample of 20 loans. Find the critical region for a test at the 10% significance level, then decide what 17 e-books would mean.

1. X ~ B(20, 0.65) under H₀. H₀: p = 0.65 and H₁: p ≠ 0.65, a 2-tail test, so each tail gets 5%.
2. Lower tail: P(X ≤ 8) = 0.0196 but P(X ≤ 9) = 0.0532, which is over 0.05. So the lower part is X ≤ 8.
3. Upper tail: P(X ≥ 17) = 0.0444 but P(X ≥ 16) = 0.1182. So the upper part is X ≥ 17.
4. The critical region is X ≤ 8 or X ≥ 17, with actual significance level 0.0196 + 0.0444 = 0.064, or 6.4%.
5. 17 lies inside the critical region, so H₀ is rejected: there is evidence at the 10% level that the proportion of e-book loans has changed.

Hypothesis test for a normal mean

If X ~ N(μ, σ²), the mean of a random sample of size n has its own normal distribution, with the same mean and a smaller variance. Edexcel's specification gives:

Example 3: battery life

A smartwatch battery lasts a time that is normally distributed with mean 36 hours and standard deviation 2.5 hours. After a software update, a random sample of 16 watches has a mean battery life of 34.9 hours. Test, at the 5% level, whether the mean has fallen. Assume the standard deviation is unchanged.

1. H₀: μ = 36 and H₁: μ < 36, a 1-tail test.
2. Under H₀, the sample mean ~ N(36, 2.5² ÷ 16), so its standard deviation is 2.5 ÷ 4 = 0.625.
3. z = (34.9 − 36) ÷ 0.625 = −1.76.
4. P(Z < −1.76) = 0.0392, which is less than 0.05. (Equivalently, −1.76 is beyond the critical value −1.6449.)
5. H₀ is rejected: the sample suggests, at the 5% level, that mean battery life has dropped since the update.

Testing for correlation

The product moment correlation coefficient, r, measures how close points lie to a straight line, from −1 to 1. To test whether the population is correlated, Edexcel says to state hypotheses in terms of ρ with H₀: ρ = 0, and compare r with a given critical value or use a given p-value.

Example 4: practice and sight-reading

For 12 piano pupils, the correlation between weekly practice time and sight-reading score is r = 0.56. Test at the 5% level whether there is positive correlation. The critical value for n = 12 at 5% (1-tail) is 0.4973.

1. H₀: ρ = 0 and H₁: ρ > 0.
2. 0.56 > 0.4973, so r is in the critical region.
3. H₀ is rejected. At the 5% level, the data support a positive correlation between practice time and sight-reading score among pupils like these.

Correlation is not proof of cause: the test only shows the variables tend to rise together. Check your answers to any of these examples in the hypothesis test calculator, and see the binomial distribution and normal distribution guides for the probabilities behind them.

Try it

Hypothesis Test Calculator

Run z, t and chi-squared tests and read off the p-value

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Common mistakes

• Finding P(X = 6) instead of P(X ≥ 6). Include every value as far out as yours, or further.
• Writing hypotheses in words or with the sample value. Use the parameter: H₀: p = 0.1, not "X = 6" or "the claim is true".
• Not halving for a 2-tail test. Each tail gets half the significance level.
• Saying H₀ is proved. If the result isn't significant, there is insufficient evidence to reject H₀. That doesn't make it true.
• Upper tail slips. P(X ≥ 6) = 1 − P(X ≤ 5), not 1 − P(X ≤ 6).
• Leaving out context. End with a sentence about the ferries, the e-books or the batteries, not just "reject H₀".

Practise hypothesis tests under time pressure on today's A-level Maths: Statistics short timed paper. The probability course covers the binomial and normal distributions these tests rely on.

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