for Continuous Prior Distributions

The purpose of the paper is to derive and discuss the properties of a system of single sampling attribute plans obtained by minimizing the average costs under the assumptions that sampling and decision costs are linear in lot size, sample size, and the number of defectives in the lot and the sample, that sampling is without replacement, and that the distribution of lot quality is a mixed binomial distribution, i.e., each lot is produced by a process in binomial control but the process average varies from lot to lot according to a frequency distribution which is assumed to be differentiable in the neighbourhood of the break-even value. It is shown that the optimum sample size is approximately a linear function of the square root of the lot size, and that the optimum acceptance number is approximately a linear function of the sample size. The accuracy of the approximation has been evaluated numerically for the beta distribution as prior. Some auxiliary tables are given with examples of applications. A short exposition of the Poisson approximation has also been given.