Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A test decision can be wrong because samples vary.
- Type I error: convincing evidence for Hₐ (H₀ rejected), but H₀ is true.
- Type II error: no convincing evidence for Hₐ (H₀ not rejected), but Hₐ is true.
- Power: the probability of correctly rejecting H₀ when it is false, for a stated alternative value.
- Reject H₀ → only a Type I error is possible. Fail to reject → only a Type II error is possible.
Key relationships
| Item | Rule |
|---|---|
| P(Type I error) | α, chosen before collecting data (often 0.01, 0.05 or 0.10) |
| Power | P(reject H₀ when a particular alternative is true) |
| P(Type II error) | β = 1 − power |
| Good design target | power ≥ 0.80, so β ≤ 0.20 |
| Larger n | power up, β down (α unchanged) |
| Smaller standard error | power up, β down |
| True value further from p₀ | power up, β down |
| Larger α | power up, β down, but Type I risk up |
| Type I error more serious | choose a small α |
| Type II error more serious | choose a larger α and/or a larger n |
Assumptions and conventions
- Each power value refers to one specific true value of the parameter; questions give it to you.
- “Others held constant”: each factor is changed one at a time.
- Error template: “The test finds (does not find) convincing evidence that [Hₐ in context], when really [truth in context].”
Mistakes to avoid
- No context in an error description.
- Only half the error: you must give both the conclusion and the true situation.
- P(Type II) = 1 − α. Wrong: it is 1 − power.
- Calling α or power the probability that a hypothesis is true.
- “Smaller α is always safer.” It raises the Type II risk.
- Naming the wrong possible error after a decision: rejection → Type I only.
- Saying a larger sample lowers α. It raises power; α is chosen.
Quick self-check
- H₀: p = 0.20, Hₐ: p < 0.20, where p is the proportion of a café’s orders that are takeaway. Describe a Type II error. (The test does not find convincing evidence that fewer than 20% of orders are takeaway, when really fewer than 20% are.)
- α = 0.05 and power = 0.83 against p = 0.12. Find P(Type II error). (1 − 0.83 = 0.17)
- A test rejects H₀ with p-value 0.02. Which error could have been made? (Type I only)
Next: practice questions.