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?