A comparison of sample allocation between simply applied standard binomial approximation, correctly applied standard binomial approximation, and improved binomial approximation

When a sample is taken from a finite population without replacement, probability is described by the hypergeometric density function. Usually a hypergeometric density function is approximated by two binomial density functions depending on the approximation condition. On the three values of the non-detection probabilities, 10%, 50%, and 80%, three kinds of binomial allocation are compared one another in terms of three verification methods which are used in the International Atomic Energy Agency(IAEA) inspection activities. The first approximation is the simply applied standard binomial approximation which is used in the IAEA. The second approximation is the correctly applied standard binomial approximation which simultaneously applies two binomial approximations comparing always whether D ³ n or D < n after sample size calculation, therefore statistically more accurate than the simply applied standard binomial approximation of the IAEA. Here D = [M/gx] is the rounded-up number of defects corresponding to a goal amount M (generally, one significant quantity), where g is the defect fraction, x is the average amount per item in same units as M, and n is the calculated sample size from a binomial approximation given a non-detection probability. The third approximation is the improved binomial approximation developed by Mr. J.L. Jaech, more accurate than the standard binomial approximations in sample size calculation.