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A coworker in the health insurance field is pulling a random sample of patient charts who have attempted suicide for a standard government report. Every random sample requires manual chart review, which typically reveals not all of the cases were correctly classified as suicide attempts and in fact were accidents or illnesses.

For example, out of a population of N=716 patients who were reported suicide attempts, she drew a random sample of 85 patients. It turns out 55 of the 85 were confirmed as true suicide attempts after manual review.

We need to report several statistics with confidence intervals from this small sample n=55 which make inferences to the larger population N. We'd like to use the finite population correction factor since N is a finite size and relatively small. Given that we kicked out 30 of the sample records as unreliable, it's reasonable to assume a similar proportion of the entire population N=716 is also unreliable. (At the very least, we know 30 of the 716 cases are not suicides.)

How can I incorporate the fact that N itself is a random variable into the finite population correction factor formula?

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For a simple random sample without replacement, the sample mean is an unbiased estimator for the population mean. Let $W_i$ be an indicator variable for inclusion (or exclusion) in the sample. Then we denote the sample mean as $\frac{1}{n_t} \sum^n_{i = 1} W_i y_i$. It can be shown that the variance of the sample mean is

$$ Var(\frac{1}{n_t} \sum^n_{i = 1} W_i y_i) = \frac{S_t^2}{n_t} \cdot (1 - \frac{n_t}{n}) $$

In this derivation, note that all variables are fixed. However, in reality, $S_t^2$, is generally unknown. In your example, while $n_t$ is known, $n$ is also unknown. However, the sample variance, $s_t^2$, is an unbiased estimator of $S_t^2$. If we also have an unbiased estimator for $n$--and it seems you do--then we have an unbiased estimate for the variance of the sample mean.

Whether $n$ is known or a random variable will change the variance of the variance of the sample mean, but won't change our estimate of the variance of the sample mean.

It should be noted that in most practical applications in survey sampling the population total is not, strictly speaking, known, but rather estimated.

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