SELECTING THE BEST ONE OF SEVERAL BINOMIAL POPULATIONS 551 



order of magnitude of the error in our large sample calculations. For ex- 

 ample, if k = 4, f/* = 0.05 and P* = 0.90, then from Table Y we find 

 that B = 1.5026 and the two expressions in (8) yield 599.54 and 601.04. 

 Hence, it would follow from (9) that n is 600 or 601 or 602. Based on an 

 investigation of the behavior of these two approximations in the case of 

 smaller P* or larger d* values, it is estimated that the true value of n is 

 601. Even if the correct value is 600 or 602 the error would be less than | 

 of 1 per cent. Fig. 6 illustrates these bounds on the Pes for ^ = 4, 

 P* = 0.50, 0.75 and 0.99. For P* ^ 0.60 the straight line approxima- 

 tion is a closer lower bound than the normal approximation. 



It is estimated that all integer entries in Tables I through IV have an 

 error of at most 1 per cent and, in particular, that all entries under 100 

 are exact. 



OTHER VALUES OF k 



In addition to the tables and graphs for k = 2, 3, 4 and 10 there are 

 also graphs (Figs. 7 through 14) on which interpolation can be carried 

 out for k = 5 through 9 and on which extrapolation can be carried out 

 for /.• = 11 through 100. By plotting n versus log k (or 7i versus A" on 

 semi-log paper) and drawing a straight (dashed) line through the values 

 of n for A' = 4 and k = 10 we obtain results which are remarkably good 

 approximations for k > 10. The solid curve in these figures connects the 

 true values obtained for k = 2, 3, 4 and 10. 



For large values of k the theoretical justification for a straight line 

 approximation is gi^•en in Appendix V. In order to check the accuracy of 

 our procedure of drawing the straight line through the values of 7i 

 computed for A- = 4 and A- = 10, we have chosen two points at A; = 101 

 for an independent computation of the probability of a correct selection. 

 For P* = 0.90, d* = 0.10 and A- = 101 the dashed line in Fig. 12 gives n 

 as approximateh" 400. To check this we computed the normal approxi- 

 mation to the probability Pes of a correct selection for the least favor- 

 able configuration in the form 



Pes ^ r F''\x + h)f{x) dx = -]r r F''\xV2 + /i)e~"' dx (10) 



where 



2d* V^ 



Vl - d*~ 



(= 4.02015 in this example) (11) 



fix) is the normal density and Fix) is its c.d.f. This was computed by a 

 method suggested by Salzer, Zucker and Capuano^ and the result was 



