I've figured it out. It ***is*** "obvious" why Evert's procedure works.
Once you hold your head at the correct angle, as my first year calculus
lecturer
used to say.
The binom.test() confidence interval gives you the value of a random
variable
say "U" (for "upper") such that
Pr(U < p) = p0
where U is a function of the observed binomial random variable, say U =
h(X).
The observed value of U is h(x), where x is the observed value of X.
Now we want p such that Pr(X <= x) = p0 where X ~ Binom(N,p).
But when X ~ Binom(N,p),
Pr(U <= p) = p0, i.e
Pr(h(X) <= p) = p0, so if we take p = h(x) we have
Pr(h(X) <= h(x)) = p0, whence
Pr(X <= x) = p0 as desired.
Still twists my head, but.
cheers,
Rolf Turner
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