A METHOD OF SAMPLING INSPECTION 627 



2, 3, 4, etc., samples. The method of extension is somewhat compli- 

 cated although the procedure is identical in nature. For example, in 

 Minimum Double Sampling, first and second acceptance numbers and 

 corresponding first and second sample sizes must be found, and at 

 the same time the total Consumer's Risk must be properly divided 

 between the two samples. We will restrict our attention to the 

 problem of Minimum Single Sampling. 



The first seven variables defined below in Table I enter into the 

 two equations needed for the solution of the problem, the first three 

 being fixed by the requirement of the method that a definite protection 

 be provided against accepting faulty material, the fourth being fixed 

 by the requirement that the average amount of inspection shall be 

 a minimum for uniform product of process average quality. There- 

 fore, the three unknown variables are c, n, and /. The five variables 

 N, n, I, p, and pt are replaced in the solution by four variables which 

 have been obtained from the original variables by combining pt with 

 the other four, viz. M = ptN, a = ptn, z = pj, and k = p/pt. 

 Since M, P, and k are specified by the method, the unknown variables 

 are c, a, and z. Two tables showing respectively the notation ^ and 

 the disposition of the variables are presented below. 



TABLE I 



Nomenclature 



N = number of pieces in lot, 



P = Consumer's Risk, the probability of accepting a submitted lot of tolerance 



quality, 

 pt = tolerance fraction defective, 

 p = process average (expected) fraction defective, 

 c = acceptance number, the maximum allowable number of defective pieces in 



sample, 

 n — number of pieces in sample, 



/ = average (expected) number of pieces inspected per lot, 

 M = ptN = number of defective pieces in lot of tolerance quality, 

 a = ptn = expected number of defective pieces in sample drawn from lot of 



tolerance quality, 

 z = pti = product of tolerance and average (expected) number of pieces in- 

 spected per lot, 

 k = p/pt = ratio of process average to tolerance, 

 m = number of defects found in sample, 



Ciy = TTT '-rr-, = number of combinations of N thmgs taken n at a time. 



{N — n)\n\ 



The solution of the problem requires the consideration of the 



3 The symbols p and pt, as used in this problem, are assumed true parameters of 

 the universe sampled and according to the notation adopted by these Laboratories 

 should be primed, i.e. p' and pt'. For the sake of simplicity here the prime notation 

 has been omitted in the equations. Ref. W. A. Shewhart, "Quality Control," 

 Bell System Technical Journal, Vol. VI, p. 723, footnote 3, October, 1927. 



