Approximating the Upper Binomial Confidence Limit

Abstract

Abstract When a sample of size n with c "defectives" is drawn randomly from an infinite population with probability p of a defective, the upper binomial confidence limit can be approximated by two formulas based on , the upper confidence limit for the parameter m of a Poisson distribution. The approximations are easy to calculate and have precisely specified bounds on their error over a stated range of n and c/n. The error of the simpler formula is guaranteed to be within .1% (with trivial deviations) of the exact binomial confidence limit when n≥ 40 and c/n ≤ 1/10. The relative error of the second formula is not more than .1% (with no deviations) when n ≥ 20 and . Values of for small values of c are readily available from tables of the Poisson and chisquare distributions. For c ≥ 50, a simple formula permits approximating with no more than .045% relative error.