4.11 Carrying Out a Test for the Difference of Two Population Means

Setting Up the Two-Sample t-Test

To compare two population means, state hypotheses about μ1 − μ2 (H0: μ1 − μ2 = 0 vs. Ha: μ1 − μ2 ≠, <, or > 0). Check conditions: random samples/assignment, independent samples (10% condition if sampling without replacement), and normal/large sample condition (Normal population, large n≥30, or roughly symmetric data).

Computing the Test Statistic

The test statistic is t = (x̄1 − x̄2) / SE, where SE = √(s1²/n1 + s2²/n2). Degrees of freedom are usually found by calculator (Welch's approximation); use the smaller of n1−1, n2−1 as a conservative estimate by hand. Find the P-value using a t-distribution.

Making a Conclusion

Compare the P-value to α. If P-value < α, reject H0 — there is convincing evidence of a difference in means. If P-value ≥ α, fail to reject H0 — not enough evidence. Always state the conclusion in context, referencing the populations and direction of the alternative hypothesis.

Key terms

Two-sample t-test hypotheses

H0: μ1 − μ2 = 0; Ha: μ1 − μ2 ≠, <, or > 0, depending on the research question.

Standard error for difference of means

SE = √(s1²/n1 + s2²/n2), combining the variability from both independent samples.

Conditions for two-sample t-test

Random samples/assignment, independent samples (10% condition if applicable), and Normal/Large Sample condition met for both groups.

Degrees of freedom (two-sample t-test)

Usually calculated via Welch's approximation using technology; by hand, conservatively use the smaller of n1−1 or n2−1.

Conclusion in context

State whether you reject or fail to reject H0, referencing the specific populations and comparing means, based on the P-value and α.

Try an AP-style question

x̄₁ = 82, x̄₂ = 78, and SE = 1.6. What is the test statistic?

1. t = 4
2. t = 6.4
3. t = 2.5
4. t = 0.4