XXV] 



CXXXI 



Now suppose we fix our attention on Fathers with spans between 66" and 67", 

 i.e. put x = 66" '5, and let us suppose samples taken of size 19. Th-n 

 a*g x = -4799,0784 {1 - -1052,6316 + -4775,9544) 



= -6585,9302, 

 and (T 9x = -81 1,5374. 



For x = 66"-5, we have # = 68"'96 



from the regression line. 



Now we will suppose the regression line for the sample of 19(!) has been found 

 and gives for the mean span of Sons of Fathers of 66" to 67" span the value 

 y x = G8"-26. The parent population gives 68"'96. Is this a reasonable difference ? 



The distribution of y x y will be given by the curve 



andwehave 



or #'* = -74401. 



a;' 2 -74401 



Thus #= = , -, -r = '04443. 



16-74401 



We have accordingly to interpolate from our tables for x = "04443 in the column 

 n = 19. This for accuracy must be done by aid of Table XXV bis. 



We have u = 1-505,3176, S a t/ =15731, 8 4 = 27, 



%= 1-468,4491, S 2 M! = 15060, 5*^ = 27. 

 Clearly we need not use 8 4 's. 



<9 = -443, = -557, 00 = '041,1252, 

 u e = 1-488,98485+ - -000,19010 = 1-488,7948. 

 Thus ^=1-488,7948, 



and P x = -5 + V'04443 x 1-488,7948 



= 813,8145. 



Hence assuming the sample to lie within the range 0" - 7 from the value 68""96 for 

 Sons of Fathers having spans of 66" to 67" in the sampled population, the chance 

 of a deviation numerically as large as or larger than this =2 (1 P x ) = -372,37 10, 

 or we might expect 37 - 2 % of samples of 19 to give a worse disagreement with the 

 value in the sampled population. 



N.B. The reader will note that we are not comparing the mean of actual 

 isolated individuals in the sample with Fathers having spans between 66" and 67", 

 but we are comparing the mean of the Sons of this array of Fathers found from 

 the regression line of the sample with the value of the same mean as given by the 

 parent population. 



r2 



