cxxxii Tables for Statisticians and Biometricians [XXV 



We can use our tables as applied to the third type of curve to test whether 

 a sample of which we know the mean and standard deviation comes from a parent 

 population of which we know the mean. 



Let the size of the sample be n, the mean and standard deviation of the sample 

 be m and s, and the mean of the parent population be M. Then, if 



x' = (m-M)/s, 

 the distribution of #' in samples of size n is given by* 



provided the parent population be normally distributed. E. S. Pearson has shown 

 the extent to which this result may still be applied in a certain range of non- 

 normal distributions f. 



It is difficult to imagine a practical case in which we know M so accurately 

 that its probable error relative to that of m is negligible, and yet do not know 

 2 the standard deviation of the parent population with corresponding accuracy. 

 If we know both M and 2 we have two independent variables m and s to compare 

 with them, and the writer of this Introduction personally much prefers in all such 

 cases the double test to the single test which involves both characters. 



Illustration (vii). Among samples of 10 from a normal population of mean variate 

 zero and standard deviation 10, a sample occurred with mean 7"0 and standard 

 deviation 14*64 {. What is the probability of such a sample occurring at a single 

 draw as judged by the present test? 



x' = j^ = -4781, and x'* = -2286. 

 The distribution curve of x' is 



(1 



and the proper transformation x = ^ /0 = "1861. 



1 + or 2 



Turning to Table XXV under n = 10 and x = "1861, we have 

 UQ = -903,2890, S 2 w = -4832, 

 w, = -909,9040, 8 2 % = - 4443, 

 = -61, = -39, 0</> = -03965, 

 u e = -907,3241,5 + -000,0549,9 

 = 907,3791. 



* This is the case really proved by " Student," Biometrika, Vol. v. pp. 7 8 ; however, the actual 

 examples he gives do not belong to this case, but indicate that he proposed a wider application of it. 

 t Biometrika, Vol. xxi. pp. 259 et seq. 

 Such a sample was one of a set of 700 samples actually drawn from a normal population. 



