Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A test asks whether the data give convincing evidence of a difference in population means. This topic covers State and Plan; Topic 4.10 covers Do and Conclude.
- Procedure: two-sample t-test for a difference between two population means. It uses t because σ₁ and σ₂ are unknown and are replaced by s₁ and s₂.
- The samples must be independent. Paired data (same individuals twice, natural pairs) use a one-sample t-test on the differences.
- Define μ₁ and μ₂ in context: population (or true) mean, response variable with units, and which population or treatment.
Key relationships
| Item | What to write |
|---|---|
| Null hypothesis | H₀: μ₁ − μ₂ = 0, or H₀: μ₁ = μ₂ |
| “Greater”, “longer”, “higher” | Hₐ: μ₁ − μ₂ > 0, or μ₁ > μ₂ |
| “Smaller”, “shorter”, “lower” | Hₐ: μ₁ − μ₂ < 0, or μ₁ < μ₂ |
| “Different”, “affects”, “changes” | Hₐ: μ₁ − μ₂ ≠ 0, or μ₁ ≠ μ₂ |
| Randomization | Two independent random samples, or a randomized experiment |
| 10% | n₁ ≤ 0.10N₁ and n₂ ≤ 0.10N₂ (sampling without replacement only) |
| Sample data | Both n ≥ 30; or populations approximately normal; or both samples free from strong skewness and outliers |
Assumptions and conventions
- Pick the order of subtraction once and keep it from the definitions to Hₐ.
- The course’s two-sample t-test does not pool: it does not assume σ₁ = σ₂.
- In an experiment with volunteers, the 10% condition is not needed, and results apply to subjects like those in the study.
- For a small sample, show the check: a dot plot description or the 1.5 × IQR fences.
Mistakes to avoid
- x̄ in the hypotheses. Use μ.
- Two-sample test on paired data. Look for a natural partner for each value.
- One-sided Hₐ for a “difference” question because one sample mean looks bigger.
- Inequality in H₀. H₀ always states equality.
- “n < 30, so it fails.” Look at both samples first.
- Combining groups: n₁ + n₂ ≥ 30, or 10% of N₁ + N₂.
- 10% condition in a randomized experiment.
Quick self-check
- “Does a new paint dry at a different rate from the old paint?” Write Hₐ. (Hₐ: μ_new − μ_old ≠ 0, two-sided)
- Samples of 18 and 40 from populations of unknown shape. What must you check? (Both sample distributions are free from strong skewness and outliers, because one sample is under 30.)
- Random samples of 45 from populations of 600 and 380 people. Is the 10% condition met? (First: 45 ≤ 60, yes. Second: 45 > 38, no, so the condition fails.)
Next: practice questions.