Estimate a difference without pooling it away

Try

Two independent samples measure a binary outcome: 90 of 150 observations succeed in group 1 and 60 of 150 in group 2. Define p1 and p2 before subtracting. The estimate p̂1−p̂2=0.6−0.4=0.2 means a 20-percentage-point difference in this chosen orientation. Reversing the groups reverses its sign.

Predict

Predict whether estimating a difference should assume the two population proportions are equal. That would impose the very relationship the interval is meant to investigate. Also predict whether two measurements on the same person satisfy an independent-groups assumption merely because they appear in separate columns.

Experiment

For these original synthetic groups, the estimated variance of the difference is 0.6·0.4/150+0.4·0.6/150=0.0032. Its square root is approximately 0.056569. Using supplied z*=1.96 gives margin about 0.110874 and interval [0.089126,0.310874].

Group	Successes	Failures	Estimated variance contribution
1	90	60	0.0016
2	60	90	0.0016

Check all four counts, then write a sentence about p1−p2. If the study was observational, the interval describes a population difference under the sampling assumptions rather than automatically a treatment effect.

Understand

For independent groups, the usual approximate interval is (p̂1−p̂2) ± z*√[p̂1(1−p̂1)/n1+p̂2(1−p̂2)/n2]. The two terms estimate separate group variances. There is no pooled probability because interval estimation does not assume p1=p2. The standard error concerns the difference of proportions, not either individual group’s count.

Check representative random sampling or an appropriate experimental design, independence within and between groups, and sufficiently large observed success and failure counts in each group. The familiar at-least-10 guideline is an approximation check. A sampling-fraction condition may support approximate independence for sampling without replacement from large populations, but clustering and pairing need their own treatment.

A difference interval is oriented: if [L,U] estimates p1−p2, then [−U,−L] estimates p2−p1. Negating endpoints without reversing their order produces a malformed interval. Name the groups in the conclusion so a positive number cannot be mistaken for the opposite comparison.

For paired binary data, such as the same people’s responses before and after, this formula generally omits important dependence. The difference variance includes covariance, and an appropriate paired categorical method uses the paired structure. Matching sample sizes does not make the groups independent. Do not apply a paired t method to two proportions simply because the word paired appears; identify the actual response and target.

Sparse data or estimates near boundaries can make simple normal difference intervals unreliable. Alternative interval methods exist, but changing methods requires a justified rule rather than clipping endpoints after an inconvenient result. Keep precision, design validity and practical relevance separate in the final interpretation.

Apply

Draw the difference interval on a line with zero marked. Explain whether the reported range supports a direction, whether it establishes a useful magnitude, and which population or study setting it concerns. Then reverse the group order and verify that your substantive statement remains unchanged despite the new signs.

Continue

Next test the specific null of equal population proportions. That null justifies a common probability estimate for the reference distribution, unlike this interval.