#StackBounty: #confidence-interval #binomial Wikipedia's text about the Clopper-Pearson interval for binomial proportions

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?

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