XXV] 



CXXX1X 



idea of the value of u v, and then proceeding to deal with the same or highly 

 correlated individuals. 



It may be of interest in regard to problems of this sort to exhibit a further 

 xaniple of the use of the (e') test as given by Neyman and Pearson on p. 206 of their 

 paper cited above. 



Illustration. A piece of work is carried out by one set of 30 workmen according 

 to Method I, and by a second set of 40 workmen according to Method II. The two 

 sets of workmen are supposed of like ability. The resulting frequencies were: 



Here for I, m n = 53*700 sees., 81 = 1'882 sees. 



II, W B = 55*175 sees., 2 = 2*072 sees. 



Now according to the test we take 



AYJ __ <WJ 



/ ''^U '**'tJ 



1849 







= "11832, 



x' z ='1342 and x = 



n = n\ + n% 1 = 69. 



This lies outside our Table XXV for n, but the probability integral is 



*(!+/(*, 34)), 



and found from the Tables of the Incomplete B-function* = '998,2199, which agrees 

 with the value '9982 given by Neyman and Pearson. Thus the odds are about 

 277 to 1 that if the two samples were from the same population x' would lie 

 outside the limits + *3662. 



It is clear that such a problem cannot be solved directly by " Student's " ratio 

 as originally given, unless we have the two samples of equal size. In any real case 

 this would be likely to occur, for to produce equal ability in the two samples, the 

 same men would probably be used for both methods. But if this were done, correlation 

 would almost certainly come in and the (e) test would be inapplicable. Hence it 

 becomes all the more important to be certain that " Student's " test can be safely 

 used, when the two populations are correlated member for member. 



* For most practical purposes, it is adequate to use here the normal curve with standard deviation 

 l/N/^3, the standard deviation of the x' curve Now *'= --3663 and l/Vn-3=-1231, therefore 

 x'/oy = 2-975, and the corresponding probability = -99853, which for most practical inference* IB u 

 able as the correct value -99822 obtained from the B-function tables. 



1 



