Confidence intervals for binomial proportions have irregular coverage over the range of possible population parameters (e.g. see Brown et al. 2001 <Link>). How can I formally and usefully describe the properties of the confidence intervals?
Say I toss a coin ten times and obtain seven heads. Are the following statements accurate?
For the Clopper-Pearson method:
The interval 0.3475–0.9333 has been generated by a method that will on at least 95% of occasions, for any true population proportion, contain the true population proportion. The long-run frequency with which this method would yield confidence intervals containing the true population proportion pertaining to this particular experiment is at least 95%.
For the Wilson's scores method:
The interval 0.3968–0.8922 has been generated by a method that will on 95% of occasions, averaged over all population proportions, contain the true population proportion. The long-run frequency with which this method would yield confidence intervals containing the true population proportion pertaining to this particular experiment may be more or less than 95%.