  Well-recommended methods of forming `confidence intervals' for a binomial
proportion give interval estimates that do not actually meet the definition of
a confidence interval, in that their coverages are sometimes lower than the
nominal confidence level. The methods are favoured because their intervals have
a shorter average length than the Clopper-Pearson (gold-standard) method, whose
intervals really are confidence intervals. Comparison of such methods is tricky
-- the best method should perhaps be the one that gives the shortest intervals
(on average), but when is the coverage of a method so poor that it should not
be classed as a means of forming confidence intervals?
  As the definition of a confidence interval is not being adhered to, another
criterion for forming interval estimates for a binomial proportion is needed.
In this paper we suggest a new criterion; methods which meet the criterion are
said to yield $\textit{locally correct confidence intervals}$. We propose a
method that yields such intervals and prove that its intervals have a shorter
average length than those of any other method that meets the criterion.
Compared with the Clopper-Pearson method, the proposed method gives intervals
with an appreciably smaller average length. The mid-$p$ method also satisfies
the new criterion and has its own optimality property.
