DOI: 10.1515/ijb-2022-0061 ISSN:

A modified rule of three for the one-sided binomial confidence interval

Lonnie Turpin, Jeanne-Claire Patin, William Jens, Morgan Turpin
  • Statistics, Probability and Uncertainty
  • General Medicine
  • Statistics and Probability


Consider the one-sided binomial confidence interval

L , 1 $\left(L,1\right)$
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 $\alpha /\sqrt{3}$
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 $\mathrm{exp}\left(-3/n\right)$
is a better lower bound when α = 0.05 and n < 1054 and that
exp 4.6569 / n $\mathrm{exp}\left(-4.6569/n\right)$
is a better bound when α = 0.01 and n < 209.

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