Sec. 6.5 TEST OF HYPOTHESIS #o(/n = Ma) 177 



The frequency distribution in Table 6.51 displays one notable con- 

 trast to that of the x's of Table 6.21, namely, the di are more variable. 

 As a matter of fact, the standard deviation of the di is greater than that 

 of the Xi by a factor of about 1.4 in this instance in which n = 10. It 

 can be shown mathematically that the factor theoretically is y/2, 

 which = 1.414, approximately; hence the empirical results of Table 

 6.51 agree quite well with the theory. 



The following theorem summarizes some of the above information 

 and makes it more precise: 



Theorem. If a very large number of pairs of independently drawn 

 samples of n observations is taken from a normal population with 

 standard deviation = a, then: 



(a) The population of differences di = Xu — x 2 i will conform 

 to the normal distribution. 



(6) The arithmetic mean of the population of di is 0. 



(c) The standard deviation of the population of di is 



<r d = o- v z/n . 



For the situation summarized in Table 6.51, o~ d = 10 V 2/10 = 4.47, 

 an amount which agrees quite well with the 4.43 shown in that table 

 as the observed standard deviation for 403 d's. 



In practice, the standard deviation, a, nearly always is unknown 

 so that an estimate must be made from the sample. When a pair of 

 samples has been taken it has been determined by mathematical 

 analysis that the best procedure to follow is this: Lump together, 

 or pool, the sums of the squares of the x t in each sample taken sepa- 

 rately and divide that sum by 2(n — 1) before taking the square 

 root. In symbols, the following is the recommended estimate of o-: 



(6.51) s = 



W) + Z(a 2 2 ) 

 2(n - 1) 



where 2(a"i 2 ) = the sum of squares of the deviations of the X's of the 

 first sample from their mean; and likewise for 2(x 2 2 ). 



When the theorem above is applied, we obtain the following formula 

 for the sampling estimate of a d \ 



/S(x! 2 ) + 2(z 2 2 ) 

 (6.52) s~ d = sV2/n = J K1 ^ \ } 



n(n — 1) 



