538 THE BELL SYSTEM TECHXICAL JOURNAL, MARCH 1957 



Brieflj' , the technique employed here is to let the experimenter decide 

 how "close" the best and second best processes can be before he is willing 

 to relax his control on the probability of a correct selection. The selection 

 of a best process will, of course, be made on the basis of the largest \ 

 observed frequency of "success"; the only remaining problem is to | 

 determine the \alue of n. Tables and graphs which cover almost all 

 practical problems in this framework are gi^•en for determining the re- 

 quired \-alue of n. In particular, tables and graphs are gi\-en for /; = 2, 

 3, 4 and 10. Graphs are also gi\'en to approximate the result for any value 

 of k up to 100. 



This problem arises in man\' widel\' different fields of endeavor; we 

 shall briefly consider two industrial applications. One application of the 

 binomial problem is to comparative yield studies. Here success corre- 

 sponds to the making of a good unit, and the goal is to select the process 

 with the highest (long-time) yield. Another application of the binomial 

 problem is to comparative life testing studies. In this case the experi- 

 menter selects a fixed time T and defines the best process as the one for 

 which the probability of any one unit surviving this time T is highest. 

 Then, of course, a successful unit is one which survives the time T. In 

 treating this as a binomial we are discarding the information contained 

 in the exact times of failure. In many cases the times of failure are either 

 unknoAvn or very inexact; in other cases it is not known how to utilize 

 the knowledge of the exact times of failure. Hence, it would be valuable 

 to know the results for the more basic binomial problem. The time T is 

 considered fixed throughout; its value is determined by non-statistical 

 considerations. The specification and the final decision of the experi- 

 menter all refer to this predetermined time T. It should be noted that 

 the experimenter cannot use information obtained from the continuation 

 of the test beyond time T since the best process for T is not necessarily 

 the best process for a longer time, say 107". This binomial type of analysis 

 has the advantage that it does not assume any particular form of the 

 life distribution. In particular, the assumption of exponential life is 

 avoided. 



The presence of a priori information changes the number of observa- 

 tions required. An alternative specification is given which is justified by 

 certain a priori information based on past experimentation. The amount 

 of saving is briefly examined. This area of utilizing a priori information 

 to reduce the number of observations required should be investigated 

 further. 



The treatment of the problem in this paper is based on the assump- 

 tion that, after experimentation is carried out, the experimenter must 



