BUSS1020 · Quantitative Business Analysis
Hypothesis Testing: Two-Sample Tests
Hypothesis Testing: Two-Sample Tests (Week 9, Berenson Ch 10.1–10.3) extends testing to comparisons between two groups. You compare two means using the pooled-variance t (equal variances), the separate-variance t (unequal variances), or the paired t (matched observations), and you compare two proportions with a pooled estimate. The central judgement is recognising the study design — independent versus paired, and whether variances can be assumed equal — because the design dictates the formula.
What this chapter covers
- 01Independent vs paired (matched) designs
- 02Pooled-variance t for two means (equal variances assumed)
- 03Separate-variance t for two means (unequal variances)
- 04Pooled variance S²ₚ and degrees of freedom
- 05Paired-difference t-test on the differences D
- 06Test for the difference between two proportions
- 07Pooled proportion p̄ for the proportion test
- 08Link between a CI for the difference and the test decision
Paired-difference t-test
- 1 markRecognise the paired design: each employee is measured twice, so analyse the DIFFERENCES D, not two independent samples.
- 2 marksHypotheses (one-tailed, right): H₀: μ_D = 0 versus H₁: μ_D > 0.
- 1 markStandard error of the mean difference = S_D/√n = 2.16/√7 = 2.16/2.6458 ≈ 0.816.
- 2 marksTest statistic t = (D̄ − 0)/(S_D/√n) = 5.0/0.816 ≈ 6.12, with df = n − 1 = 6.
- 1 markCritical value t₀.₀₅,₆ ≈ 1.943 for a right-tailed test. Since 6.12 > 1.943, reject H₀.
- 1 markConclusion: there is strong evidence at the 5% level that the training program increased weekly output.
Key terms
- Independent vs paired samples
- Independent samples come from two separate, unrelated groups; paired (matched) samples link each observation in one group to a specific observation in the other, analysed via their differences.
- Pooled-variance t-test
- A two-mean test that combines the two sample variances into one estimate S²ₚ when the population variances are assumed equal, with df = n₁ + n₂ − 2.
- Separate-variance t-test
- A two-mean test used when population variances are NOT assumed equal; it uses each sample's variance separately with Welch–Satterthwaite degrees of freedom.
- Paired-difference test
- A one-sample t-test applied to the within-pair differences D, testing whether their mean differs from zero; df = n − 1 where n is the number of pairs.
- Pooled proportion
- The combined proportion p̄ = (X₁ + X₂)/(n₁ + n₂) used in the standard error when testing whether two population proportions are equal.
Hypothesis Testing: Two-Sample Tests FAQ
How do I know whether to use a paired or an independent-samples test?
Look at how the data were collected. If each observation in one group is naturally matched to one in the other — same person before and after, twins, or the same store in two periods — it is paired and you analyse the differences. If the two groups are separate and unrelated, the samples are independent.
When do I pool the variances for two means?
Pool (use the pooled-variance t) when you can reasonably assume the two populations have equal variances; otherwise use the separate-variance t. The pooled test is more powerful when its equal-variance assumption holds.
How does a confidence interval for the difference relate to the test?
A 95% CI for μ₁ − μ₂ that contains 0 corresponds to failing to reject H₀: μ₁ = μ₂ at α = 0.05; if the interval excludes 0, you reject. The CI also shows the size and direction of the difference, which the test alone does not.
Exam move
Build a decision tree as your first move on any two-sample question: Are the samples paired or independent? If independent, are variances equal (pooled) or not (separate)? Are you comparing means or proportions? Once the branch is chosen, the formula follows. Practise spotting paired designs from the wording, since that is the most-tested judgement. Keep applying the same five-step testing ritual from Week 8, and be ready to read a difference-of-means result off both a test statistic and a confidence interval.
