Carrying Out a Two-Mean Test
Find the p-value
- Convert the two-sample $t$ statistic to a p-value from the $t$-distribution.
- One-sided $H_a$: the tail area in that direction; two-sided: double it.
- The $df$ come from technology (a calculator's two-sample $t$ routine).
- The p-value is the chance of a gap this large if the means were truly equal.
The paired t-test
- If the design is matched pairs, run a paired $t$-test on the differences.
- Compute each $d = x_1 - x_2$, then test $H_0: \mu_d = 0$ with a one-sample $t$ on the $d$'s.
- Test statistic: $t = \dfrac{\bar{d} - 0}{s_d/\sqrt{n}}$, with $df = n - 1$ ($n$ = number of pairs).
- It's just the one-sample $t$-test applied to the differences.
Make a decision
- Compare the p-value to $\alpha$:
- p $\le \alpha$ → reject $H_0$: convincing evidence the means differ.
- p $> \alpha$ → fail to reject: not convincing evidence.
- Same rule for both the two-sample and paired tests.
Conclude in context
- State the conclusion about the two population means, in context.
- "Convincing evidence the two group means differ" (if rejected), or "not convincing" (if not).
- Name the groups and what the mean measures, with units.
- For a paired test, phrase it about the mean difference.
Pick the test from the design, then compute. Matched pairs → paired $t$-test on differences ($df = n-1$, one SE from $s_d$). Independent groups → two-sample $t$-test (added variances, technology $df$). Running a two-sample test on paired data — or vice versa — gives the wrong p-value even when every number is entered correctly.
From 7.8: two-sample $t = 2$, two-sided, p-value $\approx 0.05$; $\alpha = 0.05$.
- Compare: $0.05 \le 0.05$ → reject $H_0$ (borderline).
- Conclude: convincing evidence the two group means differ.
- A paired version would instead test $H_0: \mu_d = 0$ on the differences.
Find the p-value from the $t$ statistic (doubled for two-sided) and compare to $\alpha$: reject if p $\le \alpha$, else fail to reject. Use a two-sample $t$-test for independent groups and a paired $t$-test ($t = \frac{\bar{d}}{s_d/\sqrt{n}}$) for matched pairs. State the conclusion about the two population means in context.
The t-test tail area
Two-sided doubles the tail beyond the observed t.
For a matched-pairs design, the correct test is...
Collapse pairs to differences, then a one-sample (paired) t-test.
With a two-sided p-value of 0.05 and α = 0.05, the decision is...
0.05 ≤ 0.05, so reject (borderline).
The paired t-test statistic is t = d-bar / (s_d/√n) with df = n − 1.
It's a one-sample t-test on the paired differences.
Running a two-sample test on paired data gives the correct p-value if the arithmetic is right.
Wrong procedure → wrong p-value, even with correct arithmetic.
A paired-test conclusion is phrased about...
Paired tests are about the mean of the differences.