SAMPLING INSPECTION TABLES 7 



to find a solution that will minimize the amount of inspection for uniform 

 product* of process average quality. 



In single sampling, for each combination of sample size and allowable 

 defect number, there will be a definite probability of exceeding the allowable 

 defect number for a sample drawn from uniform product of process average 

 quality. This probability is termed the Producer's Risk. It represents 

 the chance of not accepting a lot on the basis of the sample findings under 

 these postulated conditions, and for the adopted inspection procedure is 

 thus the chance of inspecting the remainder of the pieces in the lot. The 

 average (expected) amount of inspection per lot then equals the number 

 inspected in the sampled portion plus the product of the Producer's Risk 

 and the number of pieces in the remainder of the lot. This average value 

 can be found for each combination, and the desired solution is obtained by 

 choosing that combination of sample size and allowable defect number for 

 which the average amount of inspection is smallest. 



In double sampling, an entirely similar procedure is followed. Here, 

 of course, we must consider the probability of taking a second sample when 

 the first sample fails, and then the probability of failure for the second 

 sample. The overall chance of failure constitutes the Producer's Risk for 

 the complete double sampling plan. 



No distinction is made here as to who actually inspects the remainders 

 of those lots that fail to be accepted by sample. Whether the consumer 

 does this inspection, or rejects such lots and thus in effect requires the 

 producer to do it, will be considered immaterial. Interest will be centered 

 only on the total amount of inspection done, recognizing that no matter 

 which agency performs this service the cost will probably be reflected in 

 the overall cost to the consumer. 



It should be noted that, in the theoretical developments, the number of 

 defects observed in a sample is not used to "estimate" the quality of the 

 lot. Instead, it serves to indicate what action should be taken — whether 

 the lot should be accepted, subjected to further sampling, or inspected 

 completely — the entire process constituting a set of operations which 

 when repeated over and over again produce a desired end result. 



Single Sampling — Lot Quality Protection 



The solution for this plan was given in the earlier paper,^ but will be 

 reviewed briefly since certain of the principles and terms employed will 

 be extended to the other three inspection plans. 



* By "uniform product" is meant one produced under statistically controlled con- 

 ditions such that the probabihty of producing a defective piece remains constant at 

 some definite value p. The solution thus provides for a minimum of inspection if qual- 

 it}- is statistically controlled at a per cent defective level equal to the process average 

 per cent defective. 



