Track 8 · Hypothesis Testing · Lesson 7 of 14

Two-sample t-test

Compare the means of two independent groups with Welch's t-test, report the difference with a confidence interval, and know when the pooled version applies.

35 minIntermediate7 graded exercisesMembers

A courier tried new routing software and wants to compare delivery times under the old and new systems, using separate samples of parcels from before and after the change. Do the two samples show a real difference in the mean, or could sampling variation explain the gap? This is the most common comparison in applied statistics.

Two samples are independent when the observations in one group tell you nothing about specific observations in the other: different parcels, different customers, different patients randomly assigned to two arms. The parameter is the difference of population means, $\mu_1 - \mu_2$ , usually tested against $H_0: \mu_1 - \mu_2 = 0$ .

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It has 7 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Compute Welch's t statistic, standard error and degrees of freedom for two independent samples
Prefer Welch's test to the pooled test unless equal variances are justified by design
Report the difference in means with its confidence interval and an effect size
Separate “is there a difference?” from “what causes it?” in observational comparisons

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Formula sheet

Welch's two-sample t-test (the default in the lab, R and most modern software)
$t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}, \qquad df_{W} = \frac{\left(\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}\right)^2}{\dfrac{(s_1^2/n_1)^2}{n_1 - 1} + \dfrac{(s_2^2/n_2)^2}{n_2 - 1}}$

Reference

Confidence intervals for a difference in means (Welch, pooled, paired)
Intervals for μ₁ − μ₂ from independent samples (Welch by default, pooled when variances are assumed equal) or for the mean difference of paired data.
Two-sample t-test (Welch and pooled)
Compares the means of two independent groups; Welch's version (the engine default) does not assume equal variances, the pooled version does.

Working method

Read the question twice: what is being asked, about which group or variable?
Name the method before calculating.
Keep full precision; round only the final answer.
Ask whether the answer is plausible in context.