32 BELL SYSTEM TECHNICAL JOURNAL 



of values of A' which essentially affects the value of the denominator 

 of (1). This assumption may seem at first glance rather arbitrary 

 and wide of the mark, especially since the range of values which 

 essentially affects the value of the denominator in (1) depends on the 

 value of c obtained from the sample. However, if the sample is 

 reasonably large, consisting of 100 items or more, and the proportion 

 of defectives observed is small, say 10 per cent or under, the probability 

 that universes having a proportion of defectives widely different 

 from the one observed would yield such results is so small that a 

 wide range of assumptions concerning the a priori probability of 

 such universes existing makes very little change in the final result. 



Applying, then, this assumption analytically to the basic formula 

 (1) we obtain the simpler formulae 



M /x\ /TV - X\ ''^'- /X\/ N - X 



W(X X) ^^vAwln-rj J-,\c)[n-c^ 



T^(^i,^2) -x=N-n-,c,x\/N~X\ ~ /7V+ n ^'^ 



c j\ n — c ) \ n + \ 



and by means of a transformation outlined in the Appendix w^e obtain 

 from (2), 



.Yi\/A^+1-Xi\ _ /Xo + lW N-X, 

 t A n+\-t I V / l\n^\-t 



E 



W{X,,X,)^'-^^ 



iV+1 

 w + 1 



(2a) 



Formula (2a) is the one embodied in the paper referred to in footnote 

 1. While apparently less simple than (2), it is actually easier to 

 compute when c is less than the range Xo — Xi. 



When in (2a) we set Xi = c and X2 = X the resulting formula 



(p /X + 1\/ N-X \ 



W{c, X,n,N) = \- '-^^ ^nIi^ • (^) 



71 + 1 



which is at the basis of our computational work, shows explicitly 

 certain properties which are not apparent in (2). Various analytical 

 transformations and approximations based on this formula lead to 

 several interesting extensions which are discussed in the Appendix. 

 We shall leave these phases of the problem for the present, however, 

 and discuss the results of the calculations which have been made as 

 presented on the attached charts. 



