Short on time? This page is the recap. For explanations, figures and worked examples, use the full study guide.
Recap
- The null distribution is the distribution of the test statistic when H₀ is true. Get it by simulation, or use the standard normal model when the conditions are met.
- The p-value is the probability, assuming H₀ is true, of a test statistic as extreme as the observed one or more extreme, in the direction of Hₐ.
- Small p-value: the result would be unusual if H₀ were true, so it is evidence for Hₐ. Smaller means more convincing.
- Not small: the result is not unusual under H₀. There is no convincing evidence for Hₐ, but this is not evidence that H₀ is true.
Key relationships
| Alternative | Normal model (observed z) | Simulation (observed statistic) |
|---|---|---|
| Hₐ: p > p₀ | P(Z ≥ z) | proportion of simulated values at or above it |
| Hₐ: p < p₀ | P(Z ≤ z) | proportion of simulated values at or below it |
| Hₐ: p ≠ p₀ | P(Z ≤ −|z|) + P(Z ≥ |z|) = 2P(Z ≥ |z|) | proportion at least as far from p₀ as it, on either side |
Interpretation template: “Assuming the true proportion of [population] who [response] is [p₀], there is a [p-value] probability of getting a sample proportion of [p̂] or [more extreme, in the direction of Hₐ] in a random sample of [n], by chance alone.”
Assumptions and conventions
- The normal-model p-value needs the conditions from Topic 3.5 (random, 10%, np₀ and n(1 − p₀) at least 10).
- A simulated p-value is an estimate; another simulation gives a slightly different value.
- “At least as extreme” includes the observed value itself.
Mistakes to avoid
- No assumption in the interpretation. Start with “Assuming the true proportion … is p₀”.
- “The probability that H₀ is true”: the p-value is not this.
- Wrong tail: follow the inequality in Hₐ, not the sign of z.
- One tail only for a two-sided test.
- “A large p-value proves H₀”: say the evidence is not convincing instead.
- “1 − p-value = probability Hₐ is true”: no.
Quick self-check
- Hₐ: p < p₀ and z = −0.85. Find the p-value. (P(Z ≤ −0.85) ≈ 0.1977)
- Hₐ: p ≠ p₀ and z = 2.05. Find the p-value. (2 × P(Z ≥ 2.05) ≈ 0.0404)
- Hₐ: p > p₀. In 250 simulated samples under H₀, 9 have p̂ at or above the observed value. Estimate the p-value. (9 ÷ 250 = 0.036)
Next: practice questions.