Carrying Out a Two-Mean Test
Đã giới thiệuA rounded p-value cannot settle a boundary
- A display reports p=0.05 to two decimal places. The underlying result might be 0.047 or 0.053, giving different decisions at alpha=0.05.
- Use enough precision before comparing with alpha. A rounded boundary value is not necessarily exact equality; t=2 alone is also insufficient without degrees of freedom and the alternative.
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 the smaller tail area for the symmetric null distribution.
- The $df$ come from technology (a calculator's two-sample $t$ routine). Under the null model, it is the probability of a test statistic at least as extreme in the direction or directions specified by the alternative.
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.
A paired t-test has twelve complete pairs. What are its degrees of freedom?
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The analysis has twelve differences, so df = 12 - 1 = 11, not 24 - 1.
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.
For a matched-pairs design, the correct test is...
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Collapse pairs to differences, then a one-sample (paired) t-test.
With an unrounded two-sided p-value exactly 0.05 and alpha exactly 0.05, which decision follows the stated p ≤ alpha rule?
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Under the stated exact boundary convention, reject. A rounded display of 0.05 could instead represent a value above or below alpha.
The paired t-test statistic is t = d-bar / (s_d/√n) with df = n − 1.
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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.
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Wrong procedure → wrong p-value, even with correct arithmetic.
A paired-test conclusion is phrased about...
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Paired tests are about the mean of the differences.
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.
A valid two-sided test reports p = 0.053 at alpha = 0.05. Which should a report include?
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Non-rejection is a threshold decision, not proof of no difference. Report the estimate and how the evidence was obtained.
Suppose technology reports a two-sided p-value of $0.0473$ for a valid two-sample test; $\alpha=0.05$.
- Compare: $0.0473<0.05$ → reject $H_0$. Do not decide from a two-decimal display of $0.05$.
- Conclude: convincing evidence the two group means differ.
- A paired version would instead test $H_0: \mu_d = 0$ on the differences.
Carry the reasoning to a new case
- For independent samples use the two-sample procedure; for matched measurements analyse differences.
- Keep the design and alternative fixed before the significance decision.
Match each unrounded p-value to the decision at alpha=0.05.
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Use the unrounded value or sufficient precision. Failure to reject does not prove equality.
Find the p-value from the $t$ statistic (twice the smaller tail area for a two-sided test) 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.
Write from 7 degrees of freedom
- Eight paired differences are analysed with a t test.
- The lesson degrees of freedom 11 would be 12 pairs.
What are the degrees of freedom?
Pairs, not people counted twice, set the degrees of freedom.
What are the degrees of freedom for 8 pairs?
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8 minus 1 is 7.