Interval for a Difference of Means
| English | Chinese | Pinyin |
|---|---|---|
| paired | 配对 | pèi duì |
| independent samples | 独立样本 | dú lì yàng běn |
Independent or paired?
- Comparing two means starts with one crucial question about the design.
- Independent samples 独立样本: two separate groups (e.g. treatment vs. control).
- Paired (matched-pairs) 配对: two measurements on the same or matched subjects (before/after).
- The design decides which procedure you use — get it right first.
The two-sample interval
- For independent samples, use a two-sample $t$-interval for $\mu_1 - \mu_2$:
-
$$(\bar{x}_1 - \bar{x}_2) \pm t^{*}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$
- The variances add under the root (each group's $\frac{s^2}{n}$).
- Check random, 10%, and normal/large sample for both groups.
The paired interval
- For paired data, first compute each pair's difference $d = x_1 - x_2$.
- Then run a one-sample $t$-interval on the differences:
-
$$\bar{d} \pm t^{*}\frac{s_d}{\sqrt{n}}$$
- $n$ is the number of pairs; $df = n - 1$.
Match the method to the design
- Paired data analyzed as two independent samples → wrong standard error, wrong answer.
- Independent data forced into a paired test → also wrong.
- The tell: is each value in group $1$ naturally linked to one in group $2$?
- If yes → paired; if no → two-sample.
The design dictates the procedure. If subjects are matched or measured twice (before/after), it's paired — collapse to differences and use a one-sample $t$-interval. If the two groups are separate and unlinked, it's a two-sample $t$-interval. Using the two-sample formula on paired data throws away the pairing and inflates the standard error.
Measure each runner's time before and after training.
- Same runners, two measurements → paired.
- Compute differences $d = \text{before} - \text{after}$ for each runner.
- Build a one-sample $t$-interval on the $d$'s (not a two-sample interval).
Comparing two means: independent samples use a two-sample $t$-interval $(\bar{x}_1 - \bar{x}_2) \pm t^{*}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$; paired data collapse to differences and use a one-sample $t$-interval $\bar{d} \pm t^{*}\frac{s_d}{\sqrt{n}}$. Match the method to the design, checking conditions for both groups.
Interval for μ1 − μ2
Two-sample uses added variances; paired collapses to differences.
Measuring the same runners before and after training is which design?
Same subjects measured twice → paired.
For paired data, the correct interval is...
Collapse pairs to differences, then one-sample t.
For a two-sample t-interval, the two variances are added under the square root.
SE = √(s1²/n1 + s2²/n2).
Analyzing paired data as two independent samples gives the correct standard error.
It throws away the pairing and inflates the SE.
The key question that decides the procedure is...
Linked → paired; unlinked → two-sample.