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I'm reading on the standard errors used in various hypothesis tests. For example, in tests of one population proportion, we use $\sqrt{p(1-p)/n}$.

For comparison of two population proportions, we use $$\sqrt{\hat p(1-\hat p)\bigg(\frac{1}{n_1}+\frac{1}{n_2}\bigg)}$$

where $\hat p$ is the estimate of the population proportion obtained by calculating the proportion for the combined sample.

My question is: is there a need for these standard errors to be unbiased estimates of the sampling mean standard deviation? Do we ever ensure this unbiased-ness in practice?

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In fact, for purposes of statistical inference unbiasedness of the standard deviation is not so important. The typical process for statistical inference relies instead on the consistency of the variance estimator. That is,

$$\hat\sigma^2 \overset{p}{\to} \sigma^2$$

Then given the asymptotic normality of our estimator we can use Slutzky's lemma to construct $1-\alpha$ asymptotic confidence intervals. You can see my previous answer on how this is done. Other approaches to significance tests also typically avoid the need for unbiasedness.

Indeed, it can often be difficult (if not impossible) to get unbiased variance estimates with very little gain in large-sample applications. In fact, since bias will depend on particular distributions it is not possible to find a single unbiased estimator of the variance for all possible populations. So in practice, we are often unconcerned with this quality. Of course, in small samples, it might be useful to have unbiasedness so we can have some sense of the accuracy of an estimator with some justification of why we use this estimate of accuracy. So there are some results that give unbiased estimators of the variance. If you are interested in these results I'd recommend looking at Cochran's theorem used in normal models and Bessel Corrections which can sometimes correct the bias in estimates of the population variance.

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