Hypothesis testing sits in the Statistics content of every A-Level Maths specification (Edexcel, AQA, OCR and OCR MEI), usually built around the binomial distribution in Year 12 and extended to correlation coefficients or the normal distribution in Year 13. Students often find the probability calculations manageable but lose marks on the structure around them — the wording, the comparison, and the conclusion.
This guide sets out the fixed sequence of a hypothesis test, works through a full binomial example, and covers the extensions you meet later in the course, including one-tailed versus two-tailed tests and testing correlation coefficients using the large data set.
What a hypothesis test is actually asking
A hypothesis test asks: given some sample data, is there enough evidence to say a claimed value of a parameter (like a probability p) is wrong? It never proves anything is true. It only ever concludes there is, or is not, sufficient evidence to reject a starting assumption.
The five-step structure
- Define the null hypothesis H0 and alternative hypothesis H1, in terms of the parameter, e.g. H0: p = 0.3, H1: p < 0.3.
- State the significance level given in the question (commonly 5%, 1% or 10%).
- Find the test statistic from the sample, and calculate the relevant probability under H0 (or use it to find the critical region).
- Compare the probability to the significance level (or the test statistic to the critical region).
- Write a conclusion in context, referring back to the original claim in the question.
Worked example: one-tailed binomial test
A machine is supposed to produce components with a defect rate of p = 0.2. After a change to the machine, a manager suspects the defect rate has decreased. A sample of 20 components is tested and 1 is found to be defective. Test, at the 5% significance level, whether there is evidence the defect rate has decreased.
- H0: p = 0.2, H1: p < 0.2 (a one-tailed test, since the manager suspects a decrease specifically).
- Let X be the number of defective components out of 20, so under H0, \(X \sim B(20, 0.2)\).
- Find \(P(X \le 1)\) under H0. Using the binomial cumulative distribution, \(P(X \le 1) = P(X=0) + P(X=1) \approx 0.0115 + 0.0576 = 0.0692\).
- Compare 0.0692 to the significance level 0.05. Since 0.0692 > 0.05, the result is not significant.
- Conclusion: there is insufficient evidence at the 5% level to conclude that the defect rate has decreased; the observed result is consistent with p = 0.2.
Notice the direction of the inequality in step 3: because H1 claims a decrease, you find the probability of a result 'as extreme or more extreme in that direction', which here means X ≤ 1, not X = 1 alone. Using the wrong inequality is one of the most common errors in this topic.
One-tailed vs two-tailed tests
| Wording in the question | Alternative hypothesis | Tail(s) |
|---|---|---|
| 'has decreased', 'is less than' | H1: p < value | One-tailed, lower tail |
| 'has increased', 'is more than' | H1: p > value | One-tailed, upper tail |
| 'has changed', 'is different from' | H1: p ≠ value | Two-tailed — split the significance level in half for each tail |
Critical regions
Some questions ask you to find the critical region rather than test a specific observation. The critical region is the set of values of X so extreme that, if observed, you would reject H0. You build it from the tail of the distribution, adding values until the cumulative probability first exceeds the significance level, then stopping one value earlier if needed — the actual significance level achieved is usually not exactly 5% because the binomial distribution is discrete.
- State the actual significance level of the critical region once you have found it — this is often worth its own mark, since the discrete binomial rarely lands exactly on the target percentage.
- Once the critical region is found, any later observation can be tested instantly by checking whether it falls inside it — no need to recalculate probabilities from scratch.
- Watch for asymmetric critical regions in two-tailed tests, since the binomial is not symmetric unless p = 0.5.
Testing correlation coefficients (Year 13)
In Year 13, the same five-step structure is applied to the product moment correlation coefficient (PMCC), often using data connected to the large data set. Here H0: ρ = 0 (no linear correlation in the population) and H1: ρ ≠ 0, or ρ > 0, or ρ < 0 depending on the claim. The calculated PMCC from the sample is compared against a critical value from a table, provided in the exam, rather than a probability calculated from scratch.
The reasoning is identical to the binomial case: if the sample correlation coefficient is more extreme than the critical value, reject H0 and conclude there is evidence of linear correlation in the population; otherwise, there is insufficient evidence.
Common mistakes across all versions of the test
- Writing H1 with the wrong inequality direction relative to the wording of the question.
- Calculating P(X = k) instead of the cumulative tail probability P(X ≤ k) or P(X ≥ k).
- Forgetting to halve the significance level for a two-tailed test.
- Concluding 'H0 is true' or 'H0 is accepted' rather than the correct 'insufficient evidence to reject H0' — a hypothesis test never proves the null hypothesis is true.
- Leaving the conclusion generic ('reject H0') instead of restating it in the context of the original question.
A revision checklist
- Can you state H0 and H1 correctly from a worded claim, including choosing one-tailed or two-tailed?
- Can you identify the correct binomial probability to calculate, with the inequality pointing the right way?
- Can you compare correctly against a (possibly halved) significance level?
- Can you write a conclusion in context without claiming H0 has been proven true?
- Can you find a critical region from scratch and state its actual significance level?