SELECTING THE BEST ONE OF SEVERAL BINOMIAL POPULATIONS 543 

 CONSTRUCTION OF THE TABLES 



Consider any fixed value of d*. For each of a set of increasing values 

 of n the minimum probability Pes of a correct selection for d ^ d* 

 (i.e., the probabiUty for the least favorable configuration) was computed. 

 These calculations were then inverted to find the smallest n for which 

 the Pes is greater than or equal to the specified value P*. Tables I 

 through IV give the smallest value of n for k — 2, 3, 4, and 10, for d* = 

 0.05 (0.05) 0.50, and for selected values of P*. Graphs corresponding to 

 these tables are given in Figs. 2 through 5. 



For small values of n (say, ?t < 10) it was necessary to approximate 

 P[i] by calculating the Pes exactly for several values of p[i] and proceed- 

 ing in the direction of the minimum probability Pes- For the special 

 case n = 2 and A; = 3 an explicit formula for p[i] is given on Fig. 1. 



For large values of n (say, n > 10) the Pes was calculated by assum- 

 ing the symmetric configuration. Here it was necessary to make use of 

 the normal approximation to the binomial. Fortunately the appropriate 

 table needed in this normal approximation is already published." The 

 proof that this table is appropriate is given in Appendix III. The result- 

 ing value of 71 is given by 



n^^,(l-d*')^^, (8) 



where the constant B, depending on P* and k, is equal to jC and C is the 

 entry in the appropriate column of Table I of R. E. Bechhofer's paper. 

 A short table of B values. Table V (see page 550), is included in this 

 paper to make it self-contained. 



The middle expression in (8) will be referred to as the normal ap- 

 proximation and the right hand expression in (8) will be referred to as 

 the "straight line" approximation. In many cases it has been empiri- 

 cally found that these two expressions give close lower and upper bounds 

 to the true value. Thus by noting the curves drawn in Figs. 4 and 6 for 

 k = 4, P* ^ 0.75 it appears that for all values of d* the true Pes is 

 between the normal approximation and the straight line approximation. 

 Assuming this to be so, it follows that for k = 4, P* '^ 0.75 the required 

 value of n satisfies the inequalities 



[A (1 - .-)] S n S [,4J 



(9) 



where [x] denotes the smallest integer greater than or equal to the en- 

 closed quantity x. This result (9) is empirical and not based on any 

 mathematically proven inequalities. It is used here only to estimate the 



