556 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



>J^ 



Pes = 0.9168 as compared to the value 0.90 in Fig. 12. The expression 

 (10) is derived in Appendix III. Another check was made at P* = 0.99, 

 d* = 0.20 and k = 101. The value of 71 from Fig. 14 is 162. The value of 

 the Pes computed from (10) using Salzer, Zucker and Capuano^ is 

 0.9925+. 



Further calculation using (10) yielded the more accurate results 378 

 and 154 instead of 400 and 162, respectively, in the above illustrations. 

 The error in both cases is less than 6 per cent; for smaller values of k the 

 percentage error Avill, of course, be much less. 



For interpolation the results are estimated to be within 1 per cent 

 of the correct value. For example we estimate from Fig. 11 that the re- 

 quired value of n for k = 5, P* = 0.85 and d* = 0.05 is 523. This value 

 was computed by the normal approximation and found to be 522. 



TIED POPULATIONS 



In computing the tables and graphs it was assumed that if two or more 

 populations are tied for first place then one of these is selected by a chance 

 device which assigns equal probability to each of them. The experi- 

 menter may want to select one of these contenders for first place by 

 economic or other considerations. In most practical problems we may 

 assume that such a selection is at random as far as the probability of a 

 correct selection is concerned. Hence, it appears reasonable to use the 

 tables in this paper without any corrections even when the rule for tied 

 populations is altered in the manner described above. 



It is interesting to note that in the yield problem the experimenter 

 may settle the question of ties for first place by taking more observa- 

 tions mitil the tie is broken. However, in the life-testing problem he may 

 not settle ties by letting the test run beyond time T since the best process 

 for time T is not necessarily the best for a time greater than T. 



In some applications when there are two or more populations tied for 

 first place, the experimenter may prefer to recommend all these con- 

 tenders for first place rather than select one of them by a chance device. 

 In this case we shall agree to call the selection a correct one if the recom- 

 mended set contains the best population (or, when /;[i] = 'p[2] , if the 

 recommended set contains at least one of the best populations). Exact 

 tables for the procedure so altered have not been computed. However, 

 if the value of ?i is large and this rule for tied populations is used, then 

 the experimenter may reduce the tabled Aalues by an amount ecjual to 

 the largest integer contained in l/d*. For example, using the abo^•e rule 

 for tied populations for the case k = 2, P* = 0.99, d* = 0.30, the tabled 

 value 29 can be reduced by 3 giving the result 26. 



