Every test decision can be wrong. A Type I error rejects a true null hypothesis; a Type II error fails to reject a false one. Power is the chance the test catches a real effect. See all three as areas and learn what moves them.
A test can be wrong in two ways. The gray curve shows how p̂ varies if H₀ is true; the blue curve shows how p̂ varies if the true proportion is the value you set. The red dashed line is the cutoff: p̂ values beyond it lead us to reject H₀.
Rejecting H₀ when it is true. Its probability is α, chosen before the test.
Failing to reject H₀ when it is false. Its probability is β, which depends on the true value of p.
Power = 1 − β = P(reject H₀ when a specific alternative is true). Researchers usually want power of at least 0.80.
For a fixed n, lowering α makes Type I errors rarer but Type II errors more likely. Only a larger sample lowers both.
Larger n, larger α, a true value farther from p₀ (bigger effect), and less variability.
Ask what happens after each mistake. If a false alarm is costly, choose a small α; if missing an effect is costly, aim for high power.
The four possible outcomes of a test.
| Decision | If H₀ is true | If H₀ is false |
|---|---|---|
| Reject H₀ | Type I error, probability α | Correct decision, probability = power |
| Fail to reject H₀ | Correct decision, probability 1 − α | Type II error, probability β |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 3.8. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.