Statistical　inference　about　parameters　should　depend　on　raw　data　only　through　sufficient　statistics-the　well　known　sufficiency　principle.　In　particular,　inference　should　depend　on　minimal　sufficient　statistics　if　these　are　simpler　than　the　raw　data.　In　this　article,　we　construct　one-sided　confidence　intervals　for　a　proportion　which:　(i)　depend　on　the　raw　binary　data,　and　(ii)　are　uniformly　shorter　than　the　smallest　intervals　based　on　the　binomial　random　variable-a　minimal　sufficient　statistic.　In　practice,　randomized　confidence　intervals　are　seldom　used.　The　proposed　intervals　violate　the　aforementioned　principle　if　the　search　of　optimal　intervals　is　restricted　within　the　class　of　nonrandomized　confidence　intervals.　Similar　results　occur　for　other　discrete　distributions.