I'm trying to come up with a 95% confidence interval for the click-through-rate of particular advertisement. It has $x$ clicks out of $n$ impressions so far.
What's the best way to compute this, given that I expect the click-through-rate to be small? I've been told that the "usual" methods of computing a confidence interval don't do well when the true probability $p$ is near 0.
For advertisements, the true click probability is typically in the (0, 0.02) range. I don't have an exact formula for the prior, but any reasonable approximation centered in the (0, 0.02) range would do.
Is there a nice formula of something like
(lower, upper) = confidence_interval(x, n, prior_p, 0.95)
Or alternately, has anyone out there used one of the "usual" confidence interval formulas in this situation, and can confirm that it produces "close enough" results?