4.10 Setting Up a Test for the Difference of Two Population Means

Purpose of a Two-Sample t-Test

A two-sample t-test for means determines whether there is a significant difference between two population means, μ₁ and μ₂. The hypotheses are typically H₀: μ₁ − μ₂ = 0 and Hₐ: μ₁ − μ₂ ≠ 0 (or <, >). Data come from two independent random samples or randomly assigned treatment groups, not paired measurements.

Checking Conditions

Three conditions must be verified: Random (each sample is randomly selected or randomly assigned), Independent (both within and between samples — use the 10% condition for sampling without replacement), and Normal/Large Sample (both populations are approximately Normal, or both sample sizes are large enough, n₁ ≥ 30 and n₂ ≥ 30, or graphs show no strong skew/outliers).

Test Statistic and Degrees of Freedom

The test statistic is t = [(x̄₁ − x̄₂) − 0] / √(s₁²/n₁ + s₂²/n₂), using the unpooled (Welch) standard error since equal variances aren't assumed. Degrees of freedom are found using technology (a complex formula) or conservatively as the smaller of n₁ − 1 and n₂ − 1.

Key terms

Null hypothesis for two-sample t-test

H₀: μ₁ − μ₂ = 0 (no difference between the population means).

Three conditions for two-sample t-test

Random, Independent (10% condition within each sample, plus independence between samples), and Normal/Large Sample.

Test statistic formula

t = [(x̄₁ − x̄₂) − 0] / √(s₁²/n₁ + s₂²/n₂)

Why unpooled standard error is used

Because we don't assume the two population variances are equal; it's more robust than pooling.

Degrees of freedom for two-sample t-test

Calculated by technology using a complex formula, or conservatively estimated as min(n₁ − 1, n₂ − 1).

Try an AP-style question

A teacher tests whether a new method RAISES mean scores compared with the old method. Which hypotheses fit?

1. H₀: x̄_new = x̄_old; Hₐ: x̄_new > x̄_old
2. H₀: μ_new − μ_old = 0; Hₐ: μ_new − μ_old > 0
3. H₀: μ_new − μ_old > 0; Hₐ: μ_new − μ_old = 0
4. H₀: μ_new − μ_old = 0; Hₐ: μ_new − μ_old ≠ 0