Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A hypothesis test uses a random sample to decide whether there is convincing evidence against a claimed value of a parameter.
- For one population and a two-outcome variable, the procedure is the one-sample z-test for a population proportion.
- Parameter: p = the true proportion of [population] who/that [success], in context.
- H₀ is the status quo and contains “=”. Hₐ is the researcher’s suspicion and never contains “=”.
- Set-up = State (parameter, hypotheses) + Plan (name the test, check conditions).
Key relationships
| Item | Form |
|---|---|
| Null hypothesis | H₀: p = p₀ |
| One-sided alternatives | Hₐ: p > p₀ (“increased”, “more than”) or Hₐ: p < p₀ (“decreased”, “fewer than”) |
| Two-sided alternative | Hₐ: p ≠ p₀ (“different”, “changed”) |
| “At least” / “at most” claims | Test at the boundary: H₀: p = p₀ |
| Random condition | Data from a random sample of the population |
| 10% condition | n ≤ 0.10N when sampling without replacement |
| Normality condition | np₀ ≥ 10 and n(1 − p₀) ≥ 10 (expected counts under H₀) |
Assumptions and conventions
- The variable is categorical with two outcomes, and you count successes.
- p₀ comes from the claim in the question, not from the sample.
- The normal model for p̂ is centred at p₀ because the test assumes H₀ is true.
- For a confidence interval, normality used observed counts; for a test it uses expected counts.
Mistakes to avoid
- Hypotheses with p̂ instead of p.
- The researcher’s claim in H₀. It belongs in Hₐ.
- A one-sided Hₐ chosen because of the sample result. Read the question’s wording instead.
- Using np̂ in the normality check for a test.
- “n ≥ 30” or “population is normal”: not conditions for proportions.
- “SRS ✓” with no context: say what was randomly selected.
- A vague parameter: name the population and the response variable.
Quick self-check
- A café says 30% of customers order oat milk. The owner thinks the true figure has changed. Write the hypotheses. (H₀: p = 0.30; Hₐ: p ≠ 0.30, where p is the true proportion of the café’s customers who order oat milk)
- H₀: p = 0.10 with a random sample of n = 60. Is the normality condition met? (No: np₀ = 6 < 10, although n(1 − p₀) = 54 ≥ 10)
- A random sample of 400 is taken without replacement from a population of 3,000. Is the 10% condition met? (No: 0.10 × 3,000 = 300, and 400 > 300)
Next: practice questions.