Selecting the Best One of Several 

 Binomial Populations 



By MILTON SOBEL and MARILYN J. HUYETT 



(Manuscript received Jul}' 17, 1956) 



Tables have been prepared for use in any experiment designed to selec 

 that particular one of k binomial processes or popidations with the highest 

 {long time) yield or the highest probability of success. Before experimenta- 

 tion the experimenter chooses two constants d* and P* (0 < d* ^ 1; ^ 

 P* < 1) and specifies that he would like to guarantee a probability of at 

 least P* of a correct selection whenever the true difference between the long- 

 time yields associated with the best and the second best processes is at least 

 d*. The tables shoiv the smallest number of units required per process to be 

 put on test to satisfy this specification. Separate tables are given for k = 2, 

 3, 4 CLnd 10. Each table gives the result for d* = 0.05 (0.05) 0.50 and for 

 P* = 0.50, 0.60, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, and 0.99. For values of 

 d* and P* not considered in the tables, graphs are given on which inter- 

 polation can be carried out. Graphs have also been constructed to make 

 possible an interpolation or extrapolation for other values of k. An alterna- 

 tive specification is given for use when the experimenter has some a priori 

 knowledge of the processes and their probabilities of success. This specifica- 

 tion is then compared with the original specification. Applications of these 

 tables to different types of problems are considered. 



INTRODUCTION AND SUMMARY 



A frequently encountered problem is that of selecting the "best" 

 one of fc (fc ^ 2) processes or populations on the basis of the same num- 

 ber n of observations from each process. We shall assume that the given 

 processes are all binomial or "go — no go" processes and that the best 

 process is the one with the highest probability of obtaining a "success" 

 on a single observation. We shall consider a single sample or nonse- 

 (luential procedure which means that the common number n of observa- 

 tions from each process is to be determined before experimentation 

 starts. The corresponding sequential problem is being investigated.' 



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