Two-Sample T-Test

Statistics · Axiom Academy

Comparing means from two independent populations when population standard deviations are unknown 1. When to Use the Two-Sample T-Test The two-sample t-test is appropriate when you have: Two independent samples from different populations Unknown population standard deviations (σ) - the most common real-world scenario Normally distributed populations (or large sample sizes by CLT) A hypothesis about whether the population means differ Example scenarios: Comparing test scores between two teaching methods, evaluating drug effectiveness vs. placebo, or comparing salaries between two departments. When we can assume equal population variances (σ₁² = σ₂²), we combine information from both samples into a pooled variance estimate: When you have reason to believe the populations have similar spread When sample variances s₁² and s₂² are relatively close When a formal test (like Levene's test) suggests equal variances Note: The pooled variance is a weighted average that gives more weight to the larger sample. 3. The Test Statistic (Pooled) For the pooled two-sample t-test, the test statistic measures how many standard errors the sample means are apart: x̄₁ - x̄₂ is the difference in sample means s²ₚ is the pooled variance from Step 2 n₁ and n₂ are the sample sizes 4. Welch's T-Test for Unequal Variances When population variances are unequal (σ₁² ≠ σ₂²), we use Welch's t-test , which doesn't pool the variances: Key advantages of Welch's test:

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