QBUS5001 · Foundation In Data Analytics For Business
Two-Population Inference
Module 9 extends testing to two populations — the statistical engine behind A/B testing. For a difference of two means you choose among three scenarios: variances known (Z), variances unknown but equal (pooled t), and variances unknown and unequal (Welch t). You also test a difference of two proportions using a pooled estimate, compare two variances with an F-test, and check Normality with a chi-squared test.
The exam reliably tests whether you can read the scenario clues (Are variances given? Are they assumed equal?) and select the correct standard error, so scenario identification is the high-value skill here.
What this chapter covers
- 01A/B testing as a two-population comparison
- 02Two means, variances known → Z
- 03Two means, variances unknown and equal → pooled t (df = n₁+n₂−2)
- 04Two means, variances unknown and unequal → Welch t
- 05Pooled variance formula s_p²
- 06Two proportions with a pooled proportion estimate
- 07F-test for equality of two variances
- 08Chi-squared test for Normality
Two-means pooled t-test (A/B test of checkout layouts)
- 1 markHypotheses: H₀: μ₁ − μ₂ = 0 versus H₁: μ₁ − μ₂ ≠ 0 (two-tailed).
- 1 markScenario: variances unknown but equal → pooled t with df = n₁ + n₂ − 2 = 31.
- 2 marksPooled variance: s_p² = [(15−1)·8² + (18−1)·7²]/31 = (14·64 + 17·49)/31 = (896 + 833)/31 = 1729/31 = 55.77.
- 1 markStandard error: √(s_p²(1/n₁ + 1/n₂)) = √(55.77×(1/15 + 1/18)) = √(55.77×0.1222) = √6.816 = 2.611.
- 1 markTest statistic: T = (52 − 47)/2.611 = 5/2.611 = 1.915 on 31 df.
- 1 markDecision: |1.915| < t(0.025, 31) ≈ 2.040, so retain H₀ — no significant difference in mean spend between layouts.
Key terms
- A/B test
- An experiment comparing two versions (A and B) on a metric; analysed as a two-population test of means or proportions.
- Pooled variance (s_p²)
- A weighted average of two sample variances, s_p² = [(n₁−1)s₁² + (n₂−1)s₂²]/(n₁+n₂−2), used when the two populations are assumed to share a common variance.
- Welch t-test
- The two-means test used when variances are unknown and unequal; it adjusts the degrees of freedom (Welch–Satterthwaite) rather than pooling the variances.
- Pooled proportion (p̄)
- p̄ = (X₁ + X₂)/(n₁ + n₂), the combined success rate used in the standard error when testing H₀: p₁ − p₂ = 0.
- F-test for equal variances
- A test using F = s₁²/s₂² against an F distribution with (n₁−1, n₂−1) degrees of freedom to decide whether two population variances are equal.
Two-Population Inference FAQ
How do I know whether to pool the variances?
Read the question for the variance assumption. If it says variances are equal (or an F-test fails to reject equality), use the pooled t. If variances are unknown and unequal, use the Welch t. If both σ are given, use the Z version.
Why use a pooled proportion when testing two proportions?
Under H₀ the two population proportions are equal, so the best estimate of that common proportion combines both samples: p̄ = (X₁+X₂)/(n₁+n₂). This pooled value goes into the standard error of the difference.
What is the F-test actually checking?
Whether two populations have equal variances, F = s₁²/s₂². It is often used as a preliminary step to justify the equal-variance (pooled) assumption in a two-means t-test.
Exam move
Build a one-page scenario selector for two-means problems: Are both σ known? → Z. Unknown but equal? → pooled t. Unknown and unequal? → Welch t. Practise extracting the assumption clue from the wording, because the arithmetic is routine once the scenario is fixed. Keep the pooled-variance formula and the pooled-proportion formula on instant recall, since they are the only genuinely new pieces beyond Module 8.
