36 BELL SYSTEM TECHNICAL JOURNAL 



In other words, the value of W{c, X) depends mainly on the ratio 

 njN, and the values of X and c, and only in a secondary way on the 

 absolute values of n and N. This being the case, if we make a given 

 per cent sample of two different universes N and KN, the number of 

 defectives c which we may allow in our sample out of the first universe 

 N in order that our weight W may have a given value, .9 say, for the 

 true proportion of defectives in this universe to be not greater than 

 .01 is practically the same as the value of c that we may allow in the 

 sample out of the second universe KN for the same weight W and a 

 proportion of defectives .01 /i^. For values of i^ > 1 there is no 

 appreciable change introduced in the location of the c curves on 

 Charts C. For values of i^ < 1, some error is made. The magnitude 

 of this error is indicated by the cross-hatched bands on the c = 0, 

 5, 10, 15, etc., curves. The lower boundaries of these bands were 

 calculated to show the magnitude of the error introduced for the 

 corresponding values of c when K ^ A. The upper boundaries of 

 these areas correspond to values of i^ ^ 1. For other values of c 

 only the upper boundaries of the corresponding bands are shown, 

 the lower boundaries being easily deducible by visual interpolation 

 to a sufficient degree of approximation for most practical purposes. 



As examples which may serve to illustrate this sort of application 

 of Charts C consider the following: 



Example 4: A sample of w = 5,000 items has been drawn out of a 

 universe of iV = 20,000 items and c = 15 defectives were observed. 

 May we assume with a weight W = .9 or more that the true proportion 

 of defectives or trouble limit X jN is .005? 



Here .01 /X is to equal .005 for our charts to apply. Therefore, 

 K = 2. Our sample n = 500 = 2,500X and our per cent sample is 

 100 n/N =25 per cent. Corresponding then to an abscissa of 25 

 per cent and an ordinate of 2,500X on the W = .9 chart we locate a 

 point between the c = 19 and c = 20 curves. We could have allowed, 

 therefore, c = 19 defectives at the desired weight and trouble limit. 

 Since we observed a smaller number of defectives than was allowed, 

 our weight W is therefore greater than .9. As a matter of fact it is 

 practically only slightly less than .99 as appears from the W = .99 

 chart when utilized in a corresponding manner. 



Example 5: As our next example we shall attempt to determine 

 what is the trouble limit which corresponds with W = .9 to the 

 results of the sample of Example 4. On the I^ = .9 chart corre- 

 sponding to an abscissa of 25 per cent we read from the c = 15 curve 

 an ordinate of 2,015^". But this must be our sample n = 5,000. 

 We, therefore, determine K from the equation 2,015i<C = 5,000 which 

 gives K = 2.48. Hence, our corresponding trouble limit is 



