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Sampling Distributions for the Difference Between Two Sample Means

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For independent random samples, $\bar{x}_1 - \bar{x}_2$ has mean $\mu_1 - \mu_2$ and standard deviation $\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$: the variances add even though the means subtract. With $\sigma$ unknown the standard error is $\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$, which for $s_1 = 8.1$, $n_1 = 30$, $s_2 = 9.6$, $n_2 = 28$ is $\sqrt{2.187 + 3.291} \approx 2.341$. Degrees of freedom come out fractional, so either technology's value (about 53.0 here) or the conservative $\min(n_1, n_2) - 1 = 27$ is used, with the choice stated.

The spread is where this goes wrong. Variances get subtracted, producing a negative radicand of $-1.104$ for these numbers; standard errors get added, giving 3.293 instead of 2.341; or the two samples get pooled into one group of 58 and the structure disappears. The independence condition gets assumed rather than checked, so paired measurements run through a formula whose derivation requires it. And degrees of freedom get set to $n_1 + n_2 - 2$ by habit rather than to technology's value or the conservative $\min(n_1, n_2) - 1$.

method A: n=30, s=8.1. method B: n=28, s=9.6 group 1: 8.1^2 / 30 = 2.187 group 2: 9.6^2 / 28 = 3.291 ADD them: 5.478 SE = sqrt(5.478) = 2.341 the minus sign is in the DIFFERENCE, never in the SPREAD two routes that look like arithmetic subtract the variances: -1.104 no real square root add the SEs: 1.479 + 1.814 = 3.293 too large by 40%
Each group hands its own uncertainty to the comparison. The negative radicand in the first wrong route is the check that catches it before any interpretation.
two estimated SDs make df fractional option here effect technology 53.0 the closest approximation conservative min(n1,n2) - 1 27 wider interval: safe by hand not options: n1 + n2, n1 + n2 - 1, n1 + n2 - 2 by habit the two accepted values rarely change a conclusion, and whichever was used has to be stated
The conservative rule uses the smaller sample's degrees of freedom, which understates the true value and so widens the interval. Erring toward a wider interval is why it is safe to do by hand.

The work

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Lesson
Sampling Distributions for the Difference Between Two Sample Means ›

Builds the sampling distribution of a difference between two sample means: centered at the true difference, with a standard error that adds each group's own variance, plus the fractional degrees of freedom and the conservative rule that replaces them by hand.

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Diagnostic
10-item topic check ›

Ten items on two-sample spreads: variances subtracted, standard errors added, samples pooled into one group, degrees of freedom taken as n1 plus n2 minus 2, and paired designs run as two independent samples. Take it cold to find your habit, or after the lesson to check it is gone.

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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.

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