  Suppose that X_1,X_2,...,X_n are independent and identically Bernoulli(theta)
distributed. Also suppose that our aim is to find an exact confidence interval
for theta that is the intersection of a 1-\alpha/2 upper confidence interval
and a 1-\alpha/2 lower confidence interval. The Clopper-Pearson interval is the
standard such confidence interval for theta, which is widely used in practice.
We consider the randomized confidence interval of Stevens, 1950 and present
some extensions, including pseudorandomized confidence intervals. We also
consider the "data-randomized" confidence interval of Korn, 1987 and point out
some additional attractive features of this interval. We also contribute to the
discussion about the practical use of such confidence intervals.
