ELEMENTARY SAMPLING THEORY FOR ENGINEERING 35 



proportionality factor K enters also in the ratio X jN which we desig- 

 nate as the trouble limit. We shall later discuss the purpose of this 

 factor K in more detail. The understanding of the charts will be 

 simplified, however, if we consider the case for i^ = 1 in which the 

 charts become direct reading for the case of a trouble limit X jN = .01. 



The values of c, the number of defective items observed in the 

 sample, are shown as a family of curves marked c = 0, c= l,c = 2, 

 etc., sloping downward from left to right. Any point on the c = 5 

 curve, for example, on the Chart C for weight W = .9 shows the 

 corresponding values of n as ordinate and n/N as abscissa which are 

 necessary in order that this number of defectives may be accepted 

 with a degree of assurance ^ indicated by W = .9 that the true pro- 

 portion of defectives in the universe N is not greater than .01. 



It will be readily noted that for every value of the universe N, 

 there may be drawn a diagonal straight line through the origin whose 

 ordinate for an abscissa of 100 per cent sample is equal to n = N. 

 Certain representative N lines are drawn in on the charts in this 

 manner, and as many more could be inserted as desirable. Thus, 

 for a constant value of W and a constant value of X /N we have 

 provided on Charts C a ready means of determining the relationships 

 which must exist between the remaining variables N, n, and c. 



As an example of the use of these charts for the case where X = 1, 

 i.e., for X/N = .01, consider the following: 



Example 3: In a sample of w = 900 out of a universe N = 3,000, 

 what is the maximum number of defectives c that we may accept 

 with an assurance of 1^ = .9 or better that the true proportion of 

 defectives in the universe is not greater than .01? 



Referring to the Charts C for W = .9 and considering K = I, 

 we locate the point corresponding to an abscissa of 100 n/N per cent 

 = 90,000/3,000 = 30 per cent, and an ordinate n = 900. We find 

 that this lies on the diagonal straight line marked N = 3,000 K as 

 it should and that it also lies between the c = 5 and c = 6 curves. 

 From this we may infer that we may accept five defectives but not 

 six in the above case. 



We shall now proceed to explain the significance of the factor K 

 and the cross-hatched areas beneath the c = 0, 5, 10, 15, etc., curves. 

 The purpose of these features is to extend the application of Charts C 

 to values of X/N other than .01. It may be noted from the mathe- 

 matical analysis or from actual plotting of charts similar to Charts C, 

 but for different values of X/N, that the general shape and spacing 

 of the curves remains practically unchanged for any given value of W. 



■* This statement is not strictly true when we are dealing with non-integral values 

 of X. In such cases the weights W shown on the Charts C are slightly too high. 



