A note on the accuracy of an approximate interval for the binomial parameter

This paper is concerned with the problem whether the coverage probability of a well-known asymptotic confidence interval for the binomial parameter p, derived by the central limit theorem, is uniformly convergent. On the basis of several good properties this interval possesses, it seems reasonable for statisticians to conjecture that it has the uniform asymptotic confidence coefficient equal to the nominal coefficient. Surprisingly, as will be shown in this article, this conjecture is not correct for the commonly adopted nominal coefficients. Especially, this interval has a zero uniform asymptotic confidence coefficient when the nominal coefficient is less than 0.5205.