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Sampling Distributions for the Difference Between Sample Proportions

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For independent random samples, $\hat{p}_1 - \hat{p}_2$ has mean $p_1 - p_2$ and standard deviation $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$: the variances add even though the proportions subtract, so a difference is more variable than either piece. With $\hat{p}_1 = 0.46$ from 300 and $\hat{p}_2 = 0.36$ from 250, the standard deviation is about 0.042. The conditions are the one-sample three applied to each group, all four counts at least 10, plus independence between the samples, and with the counts large the difference is approximately normal.

The standard deviation is where this goes wrong. Variances get subtracted, which produces a negative quantity under the radical for these numbers; standard deviations get added directly, giving 0.059 instead of 0.042; or the two samples get pooled into one and the two-group structure disappears. Independence between the samples gets assumed rather than checked, so before-and-after measurements on the same people are run through two-sample machinery that their pairing invalidates. And the order of subtraction goes unstated, so a negative difference gets reported as though group 1 were ahead.

new system: 138 of 300. old system: 90 of 250 group 1 variance: 0.46(0.54)/300 = 0.000828 group 2 variance: 0.36(0.64)/250 = 0.000922 ADD them: 0.001750 SD = sqrt(0.001750) = 0.0418 the minus sign is in the DIFFERENCE, never in the SPREAD two routes that look like arithmetic subtract the variances: -0.0000936 no real square root add the SDs: 0.0288 + 0.0304 = 0.0591 too large by 40%
Each group hands its own uncertainty to the comparison, so the two variances add. The negative radicand in the first wrong route is the check that catches it before any interpretation.
independent samples: formula applies 300 shoppers, new system 250 DIFFERENT shoppers, old no person appears twice: variances add paired: formula does NOT apply the SAME 200 people, measured before and after the change paired measurements move together, so the difference varies LESS
Independence between the samples is the condition with no one-sample counterpart, and a design can break it silently. The same person appearing in both groups links the two proportions, and linked quantities do not obey the variances-add rule.

The work

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Lesson
Sampling Distributions for the Difference Between Sample Proportions ›

Builds the sampling distribution of a difference between two sample proportions: centered at the true difference, with a standard deviation that adds the two variances, plus the doubled conditions and the independence-between-samples requirement a paired design violates.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items on differences of proportions: variances subtracted, standard deviations added, samples pooled into one group, paired designs run as two independent samples, and subtraction order left unstated. Take it cold to find your habit, or after the lesson to check it is gone.

Not started · 10 items · ~15 min
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Targeted Practice
Drill a single misconception ›

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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