Basic Sampling - Provide confidence of estimate I tried searching the site but nothing came up.
I have a simple situation for a business problem - We have a population of 50 million files. We want to review a sample and see whether a file contains or does not contain "x". So it's binary. We will calculate a proportion of yes:no.
We want to be able to quantify how accurate our estimate is of the true population. "We are 95.. 99% confident that the true proportion is "x/y" +/- x%. From my research, confidence intervals cannot be interpreted in such a way. Rather, confidence intervals are supposed to be interpreted as "in the long-run, if we conduct many experiments, we are 95% confident that the CIs calculated for those samples will contain the true population parameter". Do we just take an arbitrary "large enough" random sample, calculate the proportion and assume that is the true proportion? I came across the binomial credible interval that can be interpreted as the former, but I know nothing of Bayesian inference unfortunately.

The short and best advice is for you to learn Bayesian inference.
The linked paper makes some very good points and gives some good recommendations, but those are not the only possible recommendations.  Rejecting any use of confidence intervals based on the paper is like "throwing out the baby with the bathwater".  There are definite problems with how some confidence intervals are interpreted, and using the wrong confidence interval procedure can lead to meaningless intervals (but using the wrong Bayesian credible interval procedure can also lead to meaningless inference).
The common approximation methods for confidence intervals for a proportion do not work very well when the true proportion is near 0 or 1.  My preferred interval in that case (and even when not near 0 or 1) is a Bayesian credible interval (or even full posterior), but you should understand what assumptions are being made and what the interval means, not just plug numbers into a formula.
You can investigate the various confidence/credible/compatibility interval procedures that work well for your likely situation and choose the one that works best (but understand what they are really saying, what assumptions are being made, and what their limitations are).
