Approximating the Lower Binomial Confidence Limit

Abstract

Abstract An earlier article [1] presented two formulas for approximating the upper binomial confidence limit from a sample of size n with c “defectives” drawn randomly from an infinite population with probability p of a defective. The present article presents two complementary formulas for approximating the lower binomial confidence limit p, based on m, the lower confidence limit for the paramater m of a Poisson distribution. These approximations are easy to calculate and have precisely specified bounds on their error over stated ranges of n and c/n. The error of the simpler formula is guaranteed to be within .1% of the exact binomial confidence limit when n ≥ 16 and c/n ≤ 1/8. The relative error of the second formula is not more than .1% when n ≥ 16 and . Values of m for small values of c are readily available from tables of the Poisson and chi-square distributions. For c ≥ 50, a simple formula permits approximating m with no more than .069% relative error. A procedure is given for approximating p when , which produces relative accuracy of at least .999 except possibly when p falls below ; then the guaranteed relative accuracy is at least .998. (Approximation of the upper confidence limit when , with relative accuracy of at least .999, is also discussed.)