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I have hypergeometric distribution with population size N. I need to estimate population proportion p. I would like to use confidence interval.

I have also three non-overlapped groups in my population. I don't need to estimate proportion for each group (stratum) but I need to do it for whole population. To make my sample more representative I consider to use stratified sampling.

Can I do stratified sampling $(n=n_1+n_2+n_3)$, then forget about strata and consider whole sample $n$? Then build CI using proportion for whole sample using standard formulas (e.g. Agresti-Coull interval)?

When I try to find the information about confidence intervals using stratified sampling, I see that I need first to calculate proportion for each stratum; then sum up weighted proportions to get estimated proportion for whole population. So all calculations are based on strata.

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  • $\begingroup$ do you need to do this analytically or can you just use software? you can easily do this with the r survey package. to answer your question, you typically compute the total variance estimator across all stratums as a linear sum of the strata estimators. $\endgroup$ Commented Jul 15, 2024 at 5:02

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With proportionate stratification, if you ignore the strata you will still get an unbiased estimate of $p$, but your confidence interval will (in general) be too wide. For binary data it's not likely to be a big issue, but we can easily construct implausibly extreme cases as an illustration.

For example, suppose $y$ is 1 always in stratum 1 and always zero in strata 2 and 3. There will be no uncertainty in the population proportion with $Y=1$ from your stratified sample, so your confidence interval will be much too long.

You can use confidence intervals for binary proportions from your favourite survey software (at least, you can if your favourite survey software is Stata or the R survey package).

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