Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- Procedure: two-sample z-test for the difference between two population proportions. Use it when a question asks for evidence of a difference between two groups’ proportions.
- Parameters: define both p₁ and p₂ as true proportions, with the response and the population (or treatment) for each.
- Hypotheses: H₀ says no difference; Hₐ says what you want evidence of.
- Conditions: randomization, 10% (sampling only), normality with p̂c.
Key relationships
| Item | Form |
|---|---|
| Null hypothesis | H₀: p₁ = p₂, or H₀: p₁ − p₂ = 0 |
| “Greater / higher / more” | Hₐ: p₁ > p₂, or p₁ − p₂ > 0 |
| “Less / lower / reduced” | Hₐ: p₁ < p₂, or p₁ − p₂ < 0 |
| “Different / affects / changes” | Hₐ: p₁ ≠ p₂, or p₁ − p₂ ≠ 0 |
| Pooled proportion | p̂c = (x₁ + x₂)/(n₁ + n₂) = (n₁p̂₁ + n₂p̂₂)/(n₁ + n₂) |
| Normality condition | n₁p̂c, n₁(1 − p̂c), n₂p̂c, n₂(1 − p̂c) all ≥ 10 |
| 10% condition | n₁ ≤ 0.10N₁ and n₂ ≤ 0.10N₂ (not needed for an experiment) |
Assumptions and conventions
- “Independent” samples: separate random samples with no individual in both and no pairing.
- In an experiment, the “populations” are the treatments: “subjects like these who would receive treatment 1”.
- Choose α and the direction of Hₐ before looking at the data.
- Interval (Topic 3.10) uses observed counts and no pooling; test uses p̂c.
Mistakes to avoid
- Hypotheses with p̂ instead of p.
- Direction of Hₐ taken from the sample results.
- Parameters defined with the samples (“of the 200 sampled”).
- Averaging p̂₁ and p̂₂ instead of pooling the counts.
- Normality checked with observed counts, or with n ≥ 30.
- 10% condition applied to an experiment, or missed for a survey.
- Paired or before-and-after data treated as independent samples.
- “Conditions are met” with no numbers.
Quick self-check
- A question asks: “Does the new timetable change the proportion of trains that run late?” Which Hₐ? (Hₐ: p_new ≠ p_old: “change” has no direction.)
- Samples: 30 of 100 and 60 of 150. Find p̂c and check normality. (p̂c = 90/250 = 0.36; counts 36, 64, 54, 96, all at least 10.)
- Is the 10% condition needed when 200 volunteers are randomly assigned to two diets? (No: it is a randomized experiment.)
Next: practice questions.