ELEMENTARY SAMPLING THEORY FOR ENGINEERING 39 



trend, improved process of manufacture, changes in personnel and so 

 on. In the final sampling of such complicated equipment as is 

 involved in a completely installed telephone central office it may be 

 necessary to take account also of breakage and such troubles due to 

 shipping and setting up of the equipment which may introduce 

 marked deviations from average conditions. 



Such general considerations as these should determine whether or 

 not we can safely assume any given value for p, and if so, what value. 

 It will be evident on a little consideration that the assumed value 

 for p need not be extremely precise for many practical applications. 



Concerning the restrictions on the function giving the a priori 

 probabilities, w{X), it may be well to point out that this function is 

 only defined for the positive integral values of X such that — X — N. 

 Moreover, since probabilities are essentially positive, it cannot be 

 negative for any of these values of X. Also since the composition of 

 the lot is certainly comprised in this range of values of X, one has 



Jlw{X) = 1. 







The questions raised concerning the form of w{X) are of particular 

 importance in connection with the economic phases of sampling, 

 that is, the relative costs of having satisfactory lots rejected and 

 unsatisfactory lots accepted by the sampling process. These costs 

 are, of course, dependent on the frequency with which given propor- 

 tions of defects occur in the lots in practice, and a detailed consideration 

 of these would itself warrant a separate treatment. 



It is felt that the general methods outlined in this treatment, while 

 not sufficiently detailed for immediate practical application to many 

 of the problems in sampling of attributes, will nevertheless serve as a 

 satisfactory basis for further work of a more specific nature. 



APPENDIX 



Case I: Assuming w{X) is a constant and noting that ^ 

 X=^n+c /x\/ N - X\ ^ / N -\- I 

 kc \cl\n-c)''\n-\-\ 



the fundamental formula 



lf%KX) 



X\/ N - X 



W{X,, X,) = ^^f-, \vw^. '^x (1) 



- N— n-rc I 



X=c \ 



X=c 

 ' Netto, "Lehrbuch der Kombinatorik," p. 15, Eq. 11 



X\IN - X 

 c i\ n — c 



