Here’s some additional material that I will add to the Wikipedia entry on Binomial Proportion Confidence Interval.
An important theoretical derivation of these confidence intervals involves the inversion of a hypothesis test. Under this formulation, the confidence interval represents those values of the population parameter that would have large p-values if they were tested as a hypothesized population.
The normal approximation interval, for example, can be represented as
<math>\left \{ \theta; -Z_{\alpha / 2} \le \frac{{\hat p - \theta}}{{\sqrt{\theta p \left ( {1-\theta} \right ) / n}}} \le Z_{\alpha / 2} \right \}</math>
This formula produces the following image:
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You can find an earlier version of this page on my old website.