Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A full test has four parts: hypotheses (with p₁ and p₂ defined in context and α stated), method and conditions, calculations, conclusion.
- The test is calculated assuming H₀: p₁ = p₂ is true, so both samples estimate one common proportion. That is why the samples are pooled.
- If H₀ is true and the conditions hold, z is approximately standard normal.
- The p-value is the probability, assuming the true proportions are equal, of a difference in sample proportions at least as extreme as the one observed, in the direction of Hₐ.
Key relationships
| Quantity | Formula or rule |
|---|---|
| Sample proportions | p̂₁ = x₁ / n₁, p̂₂ = x₂ / n₂ |
| Pooled proportion | p̂c = (x₁ + x₂) / (n₁ + n₂) |
| Test statistic | z = (p̂₁ − p̂₂ − 0) / √[p̂c(1 − p̂c)(1/n₁ + 1/n₂)] |
| p-value, Hₐ: p₁ > p₂ | P(Z ≥ z) |
| p-value, Hₐ: p₁ < p₂ | P(Z ≤ z) |
| p-value, Hₐ: p₁ ≠ p₂ | 2 × P(Z ≥ |z|) |
| Decision | p-value ≤ α: reject H₀; p-value > α: fail to reject H₀ |
Assumptions and conventions
- Conditions (from Topic 3.12): random samples or random assignment; each sample ≤ 10% of its population when sampling without replacement (not needed for an experiment); n₁p̂c, n₁(1 − p̂c), n₂p̂c, n₂(1 − p̂c) all at least 10.
- α is chosen before the data are seen; 0.05 is the usual default if none is given.
- Independent random samples support conclusions about populations; random assignment supports cause and effect for the treatments.
Mistakes to avoid
- Averaging p̂₁ and p̂₂ instead of pooling the counts (wrong unless n₁ = n₂).
- Using the unpooled standard error, which belongs to the confidence interval.
- Wrong tail, or forgetting to double for a two-sided test.
- A p-value interpretation with no assumption. Say “assuming the true proportions are equal”.
- “Accept H₀” or “the proportions are the same”.
- Conclusions about p̂₁ and p̂₂ instead of the true proportions.
- No linkage: state the p-value and α together with the decision.
Quick self-check
- Group 1 has 30 successes out of 80; group 2 has 54 out of 120. What is p̂c? (84 ÷ 200 = 0.42, not the average of 0.375 and 0.45, which is 0.4125)
- For Hₐ: p₁ > p₂, z = 1.80. What is the p-value? What would it be for Hₐ: p₁ ≠ p₂? (0.0359; 0.0719)
- A p-value is 0.0359 and α = 0.01. What is the decision? (Fail to reject H₀, because 0.0359 > 0.01. There is not convincing evidence for Hₐ.)
Next: practice questions.