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 p defined in context and α), method and conditions, calculations, conclusion.
- The method is a one-sample z-test for a population proportion.
- The test statistic z tells you how many standard errors p̂ is from p₀, assuming H₀ is true.
- If H₀ is true and the conditions hold, z has approximately a standard normal distribution (the null distribution).
- The significance level α is chosen before collecting data. It is the probability of rejecting H₀ when H₀ is true.
Key relationships
| Item | Rule |
|---|---|
| Sample proportion | p̂ = x / n |
| Test statistic | z = (p̂ − p₀) / √[p₀(1 − p₀) / n] |
| p-value, Hₐ: p > p₀ | P(Z ≥ z), right tail |
| p-value, Hₐ: p < p₀ | P(Z ≤ z), left tail |
| p-value, Hₐ: p ≠ p₀ | 2 × P(Z ≥ |z|), both tails |
| Decision | p-value ≤ α → reject H₀; p-value > α → fail to reject H₀ |
| Reject H₀ means | convincing statistical evidence for Hₐ |
| Fail to reject H₀ means | not convincing statistical evidence for Hₐ (not evidence for H₀) |
Assumptions and conventions
- Conditions (from Topic 3.5): random sample; n ≤ 10% of N when sampling without replacement; np₀ ≥ 10 and n(1 − p₀) ≥ 10.
- If no α is given, use 0.05 and say so.
- Round z to 2 decimal places and the p-value to 4. Find the p-value with a table or technology.
- Conclusion template: “Because the p-value of ___ is (less / greater) than α = ___, we (reject / fail to reject) H₀. There is (convincing / not convincing) statistical evidence that the true proportion of ___ is (less than / greater than / different from) ___.”
Mistakes to avoid
- p̂ in the standard error. Tests use p₀.
- Wrong tail. The direction comes from Hₐ, not from the sign of z.
- Not doubling for a two-sided alternative.
- “Accept H₀” or “prove”. Say “fail to reject”.
- No linkage. Name the p-value and α in the decision sentence.
- Concluding about p̂ instead of the population proportion p.
- Changing α after seeing the p-value.
Quick self-check
- n = 80, x = 36, H₀: p = 0.35, Hₐ: p > 0.35. Find z and the p-value. (p̂ = 0.45, z ≈ 1.88, p-value ≈ 0.0304)
- z = −2.10 and Hₐ: p ≠ p₀. What is the p-value? (2 × P(Z ≥ 2.10) ≈ 0.0357)
- In question 1, what is the decision at α = 0.01? (0.0304 > 0.01, so fail to reject H₀: not convincing evidence that p > 0.35)
Next: practice questions.