A METHOD OF SAMPLING INSPECTION 621 



But what proportion of the lots will fail to be accepted on the basis 

 of the sampling results? Here is where probability theory comes in 

 again. There will be a definite probability of exceeding the acceptance 

 number in samples drawn from material of process average quality. 

 Since we are interested in the amount of inspection in the long run, 

 the sample at this stage of the problem may be regarded as drawn from 

 a very large (mathematically infinite) quantity of homogeneous 

 product whose percentage defective is equal to the process average 

 per cent defective. Thus, for example, with an acceptance number 

 of 1, the average^ number of pieces inspected per lot as a result of 

 extended inspections is equal to the number of pieces in the remainder 

 of a lot multiplied by the probability of finding more than one defect 

 in a sample drawn from an infinite quantity of material of this quality. 

 This value plus the number of pieces inspected in the sample gives 

 the average amount of inspection per lot for an acceptance number of 1. 

 Similar results are found for all other acceptance numbers and the 

 desired solution is obtained by choosing that acceptance number for 

 which the average amount of inspection per lot is a minimum. 



The plan thus provides the inspector with a definite routine to 

 follow, such that his inspection effort will be a minimum under normal 



conditions. 



Charts for Single Sampling 



For any specified value of Consumer's Risk, charts may readily be 

 constructed to give the acceptance number, the sample size, and the 

 average number of pieces inspected per lot for the conditions outlined 

 above. To illustrate the general character of these charts, Figs. 2, 

 3 and 4 are presented for a Consumer's Risk value of 10 per cent. 



In the appendix it is shown that the acceptance number which 

 satisfies the condition of minimum inspection is dependent on two 

 factors, (1), the tolerance number of defects for a lot, and (2), the 

 ratio of the process average to the tolerance for defects. Fig. 2 based 

 on this relationship defines zones of acceptance numbers for which 

 the inspection is a minimum. 



Fig. 3 gives curves for finding the sample size. The mathematical 

 basis for these curves is likewise given in the appendix. For a given 

 tolerance number of defects and the acceptance number found from 

 Fig. 2, the value of tolerance times sample size is determined. This 

 quantity divided by the tolerance gives the sample size. The curves 

 shown are based on an approximation which is satisfactory for practical 

 use when the tolerance for defects does not exceed 10 per cent and the 

 sample size is not extremely small. 



2 It is to be noted that wherever "average" appears in the paper, "expected" 

 value, in the rigorous probabiHty sense, is meant. 



