Key Idea: A hypothesis test uses data to check a claim. You run a t-test or z-test, compare the p-value with α, and know the two ways the conclusion can be wrong.
Set up, choose, decide
- Set up — H₀ = no effect; H₁ = the claim. A direction means one-tailed; 'differs' means two-tailed.
- Choose — σ known → z-test; σ unknown → t-test. One sample, two samples, or paired.
- Decide — p < α → reject H₀; p ≥ α → fail to reject H₀.
- Conclude — Say what the result means in context.
| Error | Meaning | Probability |
|---|---|---|
| Type I | Reject a TRUE H₀ (false alarm) | α, from the H₀ distribution |
| Type II | Keep a FALSE H₀ (missed effect) | β, from the alternative distribution |
Lowering α makes a false alarm less likely but a missed effect more likely.
Runners are timed before and after training. Which test? A paired t-test on the differences.
p = 0.004 at the 5% level. Conclusion? Reject H₀: strong evidence the plan reduces times.
Calls ~ Po(5); reject H₀ if 10 or more calls arrive. P(Type I)? 1 − P(X ≤ 9) ≈ 0.0318.
What do you need before you can find P(Type II)? A specific alternative value of the parameter.