Hypothesis Testing
Setup
State H₀ (null) and H₁ (alternative). Choose significance level α. Compute test statistic and p-value. Reject H₀ if p < α.
Type I & II Errors
| Property | Statement |
|---|---|
| Type I (α) | Reject H₀ when it is true (false positive) |
| Type II (β) | Fail to reject H₀ when it is false (false negative) |
| Power | 1 − β |
Common Tests
| Property | Statement |
|---|---|
| z-test | Known σ, large n |
| t-test | Unknown σ, small n |
| χ²-test | Categorical data / goodness of fit |
| F-test | Comparing variances / ANOVA |
Examples
Example 1. A sample of 36 has x̄=52, s=6. Test H₀: μ=50 at α=0.05.
Solution. t = (52−50)/(6/√36) = 2. df=35, critical t≈2.03. Fail to reject H₀.
Deep Dive: Hypothesis Testing
This section builds durable understanding of hypothesis testing in statistics through definition-first reasoning, theorem mapping, and error-checking workflows.
Use a two-pass method: first derive the structure symbolically, then validate with a concrete numerical or geometric test case.
Visual Intuition
Convert algebra into a diagram, graph, or dependency map before solving. Visual-first analysis reduces sign errors and makes assumptions explicit.
Practice Set
Practice A. Re-derive one key formula on this page from first principles and annotate each transformation.
Target. Your final line should include assumptions, derivation path, and a quick verification.
Practice B. Build an application scenario using hypothesis testing and solve it with both symbolic and numeric methods.
Target. Compare outputs and explain any approximation gap.
References & Editorial Notes
- Stewart, Calculus.
- Strang, Introduction to Linear Algebra.
- Apostol, Mathematical Analysis.
Editorial update: Reviewed on 2026-04-14 for notation consistency, conceptual clarity, and exercise quality.