I'm trying to understand the following text currently (i.e., 2019-09-25) in Wikipedia about the Clopper-Pearson interval:
The Clopper–Pearson interval is an early and very common method for calculating binomial confidence intervals. This is often called an 'exact' method, because it is based on the cumulative probabilities of the binomial distribution (i.e., exactly the correct distribution rather than an approximation). However, in cases where we know the population size, the intervals may not be the smallest possible, because they include impossible proportions: for instance, for a population of size 10, an interval of [0.35, 0.65] would be too large as the true proportion cannot lie between 0.35 and 0.4, or between 0.6 and 0.65.
I do understand that in the given example it would be impossible to get an outcome that would represent a binomial proportion of 0.35 (as this would require 3.5 successes, which is not a possible outcome).
However, I believe the CP-interval is meant to represent the range of underlying probabilities of success (the 'true proportions') that have some minimum probability to produce the observed (integer) outcome. As far as I can see, these 'true proportions' can take values between 0.35 and 0.4, or between 0.6 and 0.65.
Am I seeing this wrong, or is the cited text incorrect?
Upon reflection I can see where my confusion originates from. In the context I'm working in, we have $N$ clients that have a $p$ probability that something is happening to them. Our observable is the number of clients that have actually encountered this event.
So, we are not sampling from a vase with green and red balls, and the values that $p$ can take on are fully independent of $N$ (they are definitely not integer multiples of $1/N$). I guess, the cited Wikipedia text refers to the vase case, although the introduction of the page seems to be very much applicable to my situation as well:
In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). In other words, a binomial proportion confidence interval is an interval estimate of a success probability p when only the number of experiments n and the number of successes nS are known.
Whereas I assumed Wikipedia was referring to my client group of size $N$ as the population, they were actually referring to the size of the vase.
In that light I can now understand the restrictions on the confidence interval and the discussion on the efficiency of the CP interval. Given the continuous nature of my probability value I don't think any of the restrictions of the CP method apply to my case.