212 RAYMOND PEARL AND H. M. PARSHLEY. 



in the category of estimating future expectation on the basis 

 of experience furnished by a previous sampling. Pearson (loc. 

 '/.) shows that only when a first sample is indefinitely larger 

 than a second can we with entire accuracy take the probable 

 error of the latter as .67449 1/ mpq, when p and q are the chances 

 of success and failure respectively in the former. He further 

 shows that when a first sample n gives a percentage value of the 

 character equal to p, a second sample of m individuals may be 

 expected to give 



100 p 67.440 \~pq - ~ + per cent. 



n 



In the present instance we may take the cases of service early 

 in heat as giving a "first sample" for the sex ratio in cattle, 

 and then proceed to determine whether the groups served in the 

 middle of heat and late in heat are to be regarded as random 

 samples from the "early in heat" population. 



We then have for the cases of service in middle of heat 



P -- -- 496, 



q - -504, 



m = 125, 

 n - 248, 



whence the expectation for the percentage of cf births from 

 cows served in the middle of heat, provided time or service had 

 no influence on sex, any deviation being due to errors of sampling, 

 is 49.6 3.7. Or, as a result purely of random sampling a 

 second sample from the "early in heat" population would be 

 expected to show as often as not a cf sex percentage anywhere 

 between the limits 45.9 and 53.3. 



Actually the observed percentage of males for service in 

 "middle of heat" w r as 53.6, practically the same as the upper 

 limit given. 



For cases of service in last of heat we have m = 107, and the 

 other values as before. Whence it is deduced, by the same 

 reasoning as before, that the expectation for the percentage of 

 male births from cows served late in the period of heat, provided 

 time of service had no influence on sex, any deviation being 

 due to errors of random sampling, would be 49.6 3.9. 



