Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Sections 11.1–11.4
Chapter Summary
Chapter 10 tested claims about one population; this chapter compares two. Does the new drug lower blood pressure more than the old one? Do men and women differ in average commute time? Is the proportion of satisfied customers higher this year than last? The machinery — hypotheses, P-values, confidence intervals — is the same as Chapter 10, but now the parameter is a difference: μ₁ − μ₂ or p₁ − p₂.
The chapter’s central judgment call is whether your samples are independent (two separate groups, like treatment vs. control) or dependent (paired data, like before-and-after measurements on the same people). Paired data get reduced to a single column of differences, and then it’s just a one-sample t-procedure on those differences — simpler and more powerful. For independent means you’ll meet the two-sample t-test (the Welch version, which doesn’t assume equal variances, is the safe default) and the large-sample z version. For two proportions, there’s a z-test using a pooled estimate of the common proportion. Realistic example: 200 voters surveyed before a debate (112 support a candidate) and 200 different voters after (128 support). The two-proportion z-test asks whether that 8-point bump is real or just sampling noise — with P-value ≈ 0.11, you’d conclude the data don’t give significant evidence of a change. The discipline here is matching the procedure to the design: the wrong choice (independent test on paired data) can completely miss a real effect.
Study Guide
Key terms and definitions
- Independent samples — individuals in one sample are unrelated to those in the other (e.g., two separate classes).
- Dependent (paired) samples — each observation in one sample is matched to one in the other (before/after, twins, matched pairs).
- Pooled proportion (p̄) — for testing p₁ = p₂, the combined estimate p̄ = (x₁ + x₂) ÷ (n₁ + n₂), used only in the test (not the confidence interval).
- Welch’s t-test — two-sample t procedure that does not assume equal population variances; the recommended default for independent means.
- Degrees of freedom — Welch’s formula (computed by software) or the conservative choice df = min(n₁, n₂) − 1.
Essential formulas
- Test for p₁ − p₂: z₀ = (p̂₁ − p̂₂) ÷ √[p̄(1 − p̄)(1/n₁ + 1/n₂)], with p̄ = (x₁ + x₂)/(n₁ + n₂). Requirements: independent samples, each of n₁p̂₁(1−p̂₁), n₂p̂₂(1−p̂₂) ≥ 10 (and x₁, n₁−x₁, x₂, n₂−x₂ ≥ 5 for the pooled test).
- CI for p₁ − p₂: (p̂₁ − p̂₂) ± z_{α/2} · √[p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂] (unpooled standard error).
- Test/CI for μ₁ − μ₂, independent: t₀ = [(x̄₁ − x̄₂) − Δ₀] ÷ √[s₁²/n₁ + s₂²/n₂]; CI: (x̄₁ − x̄₂) ± t_{α/2}·√[s₁²/n₁ + s₂²/n₂]. Requirements: independent random samples; each population approximately normal or each n ≥ 30.
- Paired (dependent) means: compute differences d = x₁ − x₂, then use one-sample t on the differences: t₀ = (d̄ − μ_d) ÷ (s_d/√n), df = n − 1.
- Decision rule (all cases): reject H₀ if P-value ≤ α.
Step-by-step procedures
- Choose the right procedure: (1) Are the data paired (before/after, matched)? → paired t on differences. (2) Comparing means of two separate groups? → two-sample t (Welch). (3) Comparing proportions of two groups? → two-proportion z.
- Run a paired t-test: (1) compute the differences column; (2) treat it as a one-sample t-test (Chapter 10) on the differences; (3) conclude about the mean difference.
- Run a two-proportion test: (1) compute p̂₁, p̂₂ and the pooled p̄; (2) form z₀ with the pooled SE; (3) find the P-value; (4) conclude in context.
Common mistakes to avoid
- Running an independent-samples test on paired data (or vice versa) — the design decides, not convenience.
- Using the pooled proportion in the confidence interval — pooling is for the test only; the CI uses the unpooled standard error.
- Forgetting that “no significant difference” ≠ “the two are equal” — it’s just insufficient evidence of a difference.
- Assuming equal variances to use a pooled-variance t-test without justification — Welch’s version is the safe default.
Self-check: I can…
- [ ] Distinguish independent from dependent (paired) sampling designs.
- [ ] Perform a two-proportion z-test and build a CI for p₁ − p₂.
- [ ] Perform a two-sample t-test for independent means.
- [ ] Perform a paired t-test by analyzing differences.
- [ ] Explain when pooling is and isn’t appropriate.
StatCrunch How-To
- Two proportions: Stat > Proportion Stats > Two Sample > With Data (two columns of successes/failures) or With Summary (enter x₁, n₁, x₂, n₂) > choose Hypothesis test or Confidence interval > Compute.
- Two independent means: Stat > T Stats > Two Sample > With Data or With Summary > enter both samples’ stats > Compute. (Use Stat > Z Stats > Two Sample only when both σ’s are known — rare in practice.)
- Paired means: Stat > T Stats > Paired — select the two columns; StatCrunch computes the differences and runs the test. (Equivalently: create the differences column with Data > Compute > Expression, then run Stat > T Stats > One Sample on it.)
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