ELEMENTARY SAMPLING THEORY FOR ENGINEERING 37 



-K=2TS= •^^^^- 



So far our values of K have been greater than unity, so we have not 

 had to consider our cross-hatched bands at all. In our next example 

 we shall remedy this defect. 



Example 6: What number of defective items c may be allowed in a 

 sample of 200 items out of a universe of 500 items so that W ^ .9 

 corresponding to a trouble limit X/N = .08. Here .01 /K = .08 

 .•. K = .125. Our ordinate, therefore, is 200 = l,600i^ and our 

 abscissa is 200/500 X 100 = 40 per cent sample. The point corre- 

 sponding to this on the W = .9 chart lies just below the 6=12 curve 

 indicating at first glance that we could not accept 12 defectives in 

 such a case. However, we note that K = .125 should be near the 

 lower boundary of our cross-hatched band for c = 12 if such a band 

 had been drawn in. From an inspection of the widths of the bands 

 for c = 10 and c = 15 we correctly infer that our point determined 

 by the 40 per cent sample and l,600i^ would lie well within this 

 band, and that after all we could accept 12 defectives in the example 

 in question. 



This example has been included merely to illustrate the interpre- 

 tation of the bands shown on Charts C. It may be anticipated that 

 in many if not most of the practical engineering problems only the 

 upper boundaries of the bands need be used to obtain a degree of 

 accuracy commensurate with the precision of the results desired and 

 the applicability of the basic assumptions concerning randomness 

 and the form of the a priori existence probability w{X). 



If it should be desired to extend the range of these charts to cover 

 values of W other than those shown, this may be done by means of 

 the methods outlined in the mathematical analysis, the particular 

 method to be used depending on the degree of precision required. 



The preceding pages have contained an outline of some of the 

 theory and results based on the assumption that, within a range at 

 least, all possible values of X, the unknown number of defectives, 

 were a priori, that is, before the sample in question was made, equally 

 likely. This assumption we mentioned as appropi"iate to consider in 

 case we have no information to the contrary. The results may be 

 also applicable to certain cases where we do have some information 

 of a general sort, but which it is difficult to express analytically. 

 However, it is by no means the only reasonable assumption to make 

 concerning the form of w{X) as it enters into the basic formula (1). 



