OCR section 2.05 puts a claim on trial. You set up H₀ and H₁, compute a test statistic, compare it with a critical value or p-value, and reach a conclusion in context. Tests: the binomial proportion, the Normal mean, and correlation.
The final mark is always for the conclusion in context — never just 'reject H₀'. Work through each screen, answer the questions as you go and collect ⭐ stars. Press Start when you're ready.
Two-tailed tests split the significance level. A two-tailed test at 5% puts 2.5% in each tail. Forgetting to halve it is the single most common error in this topic.
A treatment is claimed to work on 25% of patients. A researcher believes it works on more. In a trial of 20 patients, 9 improve. Test at the 5% level.
H₀: p = 0.25 H₁: p > 0.25 (one-tailed)
Under H₀, X ~ B(20, 0.25).
p-value = P(X ≥ 9) = 1 − P(X ≤ 8) = 1 − 0.9591 = 0.0409.
0.0409 < 0.05, so reject H₀.
Conclusion in context: there is significant evidence at the 5% level that the treatment works on more than 25% of patients.
Note the direction. H₁ says ‘more’, so the extreme results are the large ones — hence P(X ≥ 9), not P(X = 9) and not P(X ≤ 9). If H₁ were p < 0.25 you would compute P(X ≤ 9) instead.
Instead of a p-value, you can find the critical region in advance — the values of the test statistic that would make you reject H₀.
Because the binomial is discrete, you can rarely hit the significance level exactly. The actual significance level is the true probability of landing in the critical region when H₀ is true.
2.5% is wanted in each tail. Under H₀, each outcome has probability 10Cr/1024.
Lower tail: P(X ≤ 1) = (1 + 10)/1024 = 11/1024 = 0.01074. (P(X ≤ 2) = 56/1024 = 0.0547, which is too big — so the lower critical region is X ≤ 1.)
Upper tail: by symmetry, P(X ≥ 9) = 0.01074.
Actual significance level = 0.01074 + 0.01074 = 0.0215, i.e. 2.15% — noticeably less than the nominal 5%.
Why it matters: the discreteness makes the test more conservative than advertised. The actual significance level is the real probability of a Type I error.
If X ~ N(μ, σ²), then the sample mean of n observations has a smaller spread:
X ~ N(μ, 6²). A sample of n = 25 has mean x̄ = 52.4. Test H₀: μ = 50 against H₁: μ > 50 at the 5% level.
Standard error = σ/√n = 6/√25 = 6/5 = 1.2.
z = (52.4 − 50) / 1.2 = 2.4 / 1.2 = 2.
Method 1 — critical value: for a one-tailed 5% test, zcrit = 1.645. Since 2 > 1.645, reject H₀.
Method 2 — p-value: P(Z > 2) = 1 − Φ(2) = 1 − 0.97725 = 0.0228. Since 0.0228 < 0.05, reject H₀. Both agree. ✓
In context: there is significant evidence at the 5% level that the mean has increased above 50.
The critical values to know: one-tailed 5% ⇒ 1.645; one-tailed 1% ⇒ 2.326; two-tailed 5% ⇒ ±1.96; two-tailed 1% ⇒ ±2.576.
Testing correlation. You can test whether a sample correlation coefficient r is strong enough to be evidence of a real population correlation ρ.
The critical value depends on n. With a small sample, even a fairly big r can arise by chance, so the bar is high. With a large sample, a modest r can be significant. This is why the sample size must always be quoted.
Type I error. Rejecting H₀ when it is actually true — a false positive. Its probability is the (actual) significance level. Choosing 1% instead of 5% makes a Type I error less likely, but makes it harder to detect a real effect.
Tap the evidence, then tap the correct decision.
Tap a term on the left, then its definition.
Set up: H₀ states a specific value; H₁ is what you suspect; state the significance level
Decide: p-value < α ⇒ reject H₀; or compare the test statistic with the critical value
Two-tailed: split α — 5% two-tailed means 2.5% per tail, zcrit = ±1.96
Binomial test: X ~ B(n, p) under H₀; p-value = P(X ≥ x) or P(X ≤ x), matching H₁
Actual significance: the true tail probability of the discrete critical region — usually less than α
Normal mean: X̄ ~ N(μ, σ²/n), so z = (x̄ − μ)/(σ/√n)
Correlation: compare |r| with the critical value for that n; H₀ is ρ = 0
Type I error: rejecting a true H₀; its probability is the significance level
That is OCR 2.05 — and the last mark is always the conclusion in context. Press Finish to see your score.
You've worked through Hypothesis testing for OCR A-level Mathematics A. 🎉
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Next: test yourself in the Evaluate stage Confidence Quiz, then lock it in with Verify.