Consider the one-sided binomial confidence interval L , 1 containing the unknown parameter p when all n trials are successful, and the significance level α to be five or one percent. We develop two functions (one for each level) that represent approximations within α / 3 of the exact lower-bound L = α 1/ n . Both the exponential (referred to as a modified rule of three) and the logarithmic function are shown to outperform the standard rule of three L ≃ 1 - 3/ n over each of their respective ranges, that together encompass all sample sizes n ≥ 1. Specifically for the exponential, we find that exp - 3 / n is a better lower bound when α = 0.05 and n < 1054 and that exp - 4.6569 / n is a better bound when α = 0.01 and n < 209.
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