96 BELL SYSTEM TECHNICAL JOURNAL 



2nd — 99 in 100 that it is not less than 1.4 per cent. 



3rd— 98 in 100 that it lies between 1.4 per cent, and G.2.5 per cent. 



Lilcewise, considering the curves marked .06 if 1,000 observations 

 gave 6 per cent, of calls delayed, then we may bet 



1st — 99 in 100 that the unknown percentage of calls delayed is not 



greater than 8.05. 

 2nd — 99 in 100 that it is not less than 4.4 per cent. 

 3rd — 98 in 100 that it lies between 4.4 per cent, and 8.05 per cent. 



It is obvious from the shape of the curves that a few hundred obser- 

 vations do not give more than a vague idea as to the unknown per 

 cent, of calls delayed. On the other hand, the gain in accuracy 

 obtained by making more than 10,000 observations would hardly 

 justify the expense involved. The number of observations which 

 safety requires in any particular problem must be determined by the 

 conditions of the problem itself. If we are willing to take a chance 

 of 9 in 10 or 8 in 10 instead of 99 in 100 or 98 in 100, respectively, 

 the curves of Fig. 2 will give us an idea of the range within which 

 the unknown percentage of defectives lies. 



APPENDIX 

 Case No. 1 — Infinite Source of S.vmples 



An inspection of ;; samples has given c defecti\es. The observed 

 frequency is then c/n. Let p be the unknown true frequency and pi 

 the frequency of delayed calls which has been arbitrarily chosen as 

 being the maximum permissible. 



The a posteriori probability that p>/'i is 



f'"w{x)x'il-x)'--'dx 

 / W{x)x'{l-x)'-'dx 



Jo 



re W (x) is the a priori existence probability that p=x. This 

 iiula is unmanageable if the form of W (.r) is unknown, 

 ssumc first that IF (.v) is a constant b for o<x<g, where g>pi. 

 n 



where W (x) is the 

 form I 



Assui 

 Then 



PV il-x)»-'dx 



P= !12 ; (2) 



jTV (I -x)"-' dx+f ~^.v'(l -xy-'dx 



