ECON20003 · Quantitative Methods 2
Comparing Two Population Central Locations
Comparing Two Population Central Locations is the which-test forest in miniature. You first decide whether the two samples are paired (same units measured twice) or independent (two separate groups). For paired data you use the paired t-test on the differences. For independent groups you run an F-test of equal variances first: if variances are equal use the pooled (equal-variance) t-test, and if they differ use the Welch (unequal-variance) t-test. When normality fails, the rank-based alternatives take over — the Wilcoxon signed-rank test for paired data and the Wilcoxon rank-sum / Mann-Whitney test for independent groups.
What this chapter covers
- 01Paired vs independent designs — choose the design first
- 02Paired t-test on the differences (df = n − 1)
- 03F-test of equal variances as the gatekeeper
- 04Pooled (equal-variance) t-test with S²_p and df = n₁ + n₂ − 2
- 05Welch (unequal-variance) t-test with Satterthwaite df
- 06Nonparametric: Wilcoxon signed-rank (paired) and rank-sum/Mann-Whitney (independent)
Pooled two-sample t-test (equal variances)
- 1 markState the hypotheses (two-tailed): H₀: μ₁ = μ₂ versus H₁: μ₁ ≠ μ₂.
- 1 markBecause variances are not rejected as equal and both groups are normal, use the pooled t-test with df = n₁ + n₂ − 2 = 20.
- 2 marksPool the variances: S²_p = [(n₁−1)S₁² + (n₂−1)S₂²]/(n₁+n₂−2) = [9×36 + 11×25]/20 = (324 + 275)/20 = 599/20 = 29.95.
- 2 marksStandard error: √(S²_p (1/n₁ + 1/n₂)) = √(29.95 × (0.1 + 0.08333)) = √(29.95 × 0.18333) = √5.491 = 2.343.
- 1 markTest statistic: t = (X̄₁ − X̄₂)/SE = (85 − 80)/2.343 = 5/2.343 = 2.13.
- 1 markDecision rule: the two-tailed critical value is t₀.₀₂₅,₂₀ = 2.086. Since 2.13 > 2.086, reject H₀.
- 1 markConclude in context: there is significant evidence at the 5% level that the two training methods produce different mean scores.
Key terms
- Paired vs independent samples
- Paired samples measure the same units twice (before/after, matched pairs) and are analysed on the differences; independent samples come from two separate groups. Misclassifying the design picks the wrong test and loses easy marks.
- Pooled t-test
- The equal-variance two-sample t-test. It combines both samples into one pooled variance S²_p and uses df = n₁ + n₂ − 2; valid only when the F-test does not reject equal variances and both populations are roughly normal.
- Welch t-test
- The unequal-variance two-sample t-test. It does not pool variances and uses a Satterthwaite-adjusted df; use it when the F-test rejects equal variances.
- Wilcoxon rank-sum (Mann-Whitney)
- The nonparametric alternative for two independent groups. It compares locations by ranking all observations together, making no normality assumption — used when the t-test's normality requirement fails.
Comparing Two Population Central Locations FAQ
How do I know whether to use the pooled or the Welch t-test?
Run the F-test of equal variances first. If it does not reject H₀ (variances look equal), use the pooled t-test; if it rejects (variances differ), use Welch. When in doubt, Welch is the safer default because it stays valid even when variances are unequal.
When do I drop to the rank-based tests?
When the normality assumption behind the t-tests fails — a curved Q-Q plot or a small Shapiro-Wilk p. Use the Wilcoxon signed-rank test for paired data and the Wilcoxon rank-sum / Mann-Whitney test for two independent groups.
Exam move
Draw the decision tree once and reuse it on every two-sample question: paired or independent → normal? → equal or unequal variance (run the F-test) → parametric or nonparametric. The pooled-variance and SE formulas are on the sheet, so practise substituting fast and keeping df straight.
