Statement of result for binomial confidence intervals 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%.
 A: You ask how to 'formally and usefully' present your conclusions
Formally: Your answer is an accurate summary of some of the results from Brown et al. as I understand them.  (I note you do not offer their preferred small n method).
Usefully: I wonder who you audience is.  For professional statisticians, you could state your two intervals directly with only citations to the original papers - no further exposition needed.  For an applied audience, you would surely rather pick an interval on whatever substantive grounds you (or they) have, e.g. a preference for conservative coverage or good behavior for small proportions, etc., and just present that interval alone, noting its nominal and perhaps also its actual coverage much as you do above, perhaps with a footnote to the effect that other intervals are possible.  
As it stands you offer a choice of intervals but not much explicit guidance for an applied audience to make use of that information.  In short, for that sort of audience I would suggest either more information about the implications of choice of interval.  Or less!
A: When using confidence intervals, my response is always as follows:
In repeated sampling % of intervals so constructed will contain _. Thus, we have % confidence that the true  lies within __-_.
One example is 
In repeated sampling, 95% of all intervals so constructed will contain Mu, the true population mean. Thus we have 95% confidence that the population GPA lies between 2.72 and 3.07. 
I don't see anything particularly wrong with your response, except I would imagine the last sentence should be 95%, not more or less, or at least. 
