*Bounty: 50*

*Bounty: 50*

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.[8] 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?