628 BELL SYSTEM TECHNICAL JOURNAL 



following two equations: 



z = MM, a, c, k), 



P = MM, a, c). 



(1) 

 (2) 



where /i and/2 represent symbolic functions whxh are to be determined 

 later. 



We wish to find a pair of values {c, a) which will make 2 a minimum, 



TABLE II 

 Disposition of Variables 



subject to the condition that this pair (c, a) satisfies equation (2). 

 Hence, due to the discreteness of c, pairs (c, a) satisfying (2) are 

 substituted in (1) until a minimum value of z is found. Thus, for 

 P = .10 and for given values of M and k, we read c from Fig. 2, 

 a for this value of c from Fig. 3, and the minimum value of z from 

 Fig. 4. 



Basis of Fig. 2 Giving Minimum Acceptance Numbers 



In determining the function /i involved in equation (1), the average 

 number of pieces inspected per lot, /, is treated as the dependent 

 variable. Since a sample is always taken, n pieces will be inspected 

 from every lot submitted. The number of times that the remainder 

 of the lot (N — n) will be inspected on the average is determined 

 from the expression giving the probability that more than c defective 

 pieces will be found in n. The sample is assumed to be drawn from 

 a product of which a fraction, p, is defective. The probability that c 

 or less defective pieces will be found in n pieces selected at random 

 from a product containing p fraction defective pieces is given by the 

 sum of the first c + 1 terms of the Point Binomial [(1 — p) ■}- #J". 

 Hence the average number of pieces inspected per lot is determined 

 from the relation, 



