Track 7 · Estimation and Confidence Intervals · Lesson 7 of 10

Intervals for differences in means

Build and interpret confidence intervals for a difference in means from independent samples (Welch) and from paired data.

35 minIntermediate8 graded exercisesMembers

A courier firm installs route-optimisation software and wants to know how many minutes it saves per parcel. Measured on the same 60 routes before and after, the saving is pinned down to within about a minute; measured on unrelated deliveries, the same saving is lost in the noise. How the data were collected decides which interval to use.

Many questions compare two means: treatment versus placebo, new system versus old. The parameter is the difference $\mu_1 - \mu_2$ and the natural estimate is $\bar{x}_1 - \bar{x}_2$ . The right standard error depends on the design:

Independent samples: two separate groups, no link between an observation in one group and one in the other.
Paired data: each observation in one group is matched to one in the other — the same route before and after, the same patient's two eyes, twins.

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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Build a Welch interval for a difference in means, including its degrees of freedom
Build a paired interval from within-pair differences
Choose paired or independent methods from the study design
Interpret a difference interval: direction, units, and whether 0 is plausible

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Formula sheet

The Welch interval for μ₁ − μ₂ (independent samples)
$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2,\,\nu}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}, \qquad \nu = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1 - 1} + \frac{(s_2^2/n_2)^2}{n_2 - 1}}$
The paired t interval
$\bar{d} \pm t_{\alpha/2,\,n-1}\,\frac{s_d}{\sqrt{n}}$

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.
Paired t-test
Tests the mean of within-pair differences (before/after, matched pairs) with a one-sample t-test on the differences.

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.