316 BELL SYSTEM TECHNIC?TL JOURNAL 



ohms respectively. The true probable and 99.73% errors are 23% 

 and 148% higher respectively than those calculated by customary 

 theory, as is e\'idcnt from Fig. 5. 



Discussion of Type 1: 



Examples of this type of problem are obviously so numerous that 

 further illustrations need not be given. They occur every day in 

 practically every science. We see that in such cases it is certainly 

 necessary to allow for the effect of the small size of sample. 



Problem Type 2, Determination of Error of Average Difference 

 Example 1: 



Fi\e instruments are measured for some characteristic X, first on 

 one machine and then on another, giving two sets of values Xn, X12, 

 . . .Xi5, and X21, X22, . . .x<if, respectively. Calculate the 5 differ- 

 ences Xu — X2i = :v:i, X12 — X22 = ^2, • ■ .Xii — X2o = Xo- Assume that 

 the average difference is x and the standard deviation of the differ- 

 ences is s. Assuming the two machines give the same results except 

 for random variations, what is the probability that the observed 

 difference would occur? Are we justified in the assumption that the 

 machines give the same results? 



Solution: 



The true difference is zero on this assumption. The observed 



X — Q 



difference is s= , and "Student's" tables may be used to evaluate 



s 



this probability." If this probability is very small, let us say .001 or 

 less, it may be taken as indicating that the machines do not give the 

 same results. 



Example 2: 



We wish to compare the depth of penetration obtained from two 

 different methods of preserving chestnut telephone poles. We choose 

 n poles for test. A sample from each pole is treated by one process, 

 and a sample from each pole is treated by another process. The 

 depths of penetration are measured. Are we justified in assuming the 

 two methods to gi\e significantly different results? 



" Approximate values can be obtained from the curves in Fig. 6. 



