ccxlii 



Tables for Statisticians and Biometricians 



a-d 



the Introduction to Part I of these Tables. Rough interpolation from the actual 

 distribution, taking proportional parts of a frequency block, shows that 89*3 / of 

 the observations lie between the inner pair of limits and 99*4 / o between the 

 outer, values which are in satisfactory agreement with the prediction. 



Illustration (ii). The process of testing the hypothesis that a given sample has 

 been drawn at random from a specified population, often consists in referring a 

 frequency constant calculated from the observations to the theoretical distribution 

 it would follow in repeated sampling were the hypothesis tested true. Similar 

 problems will arise in comparing two samples. 



It is known for example that if a proportion p of the individuals in a population 

 possess a certain character A, and a proportion g = 1 p do not, then the number of 

 individuals x bearing the character in a sample of n will vary in sampling according 

 to the terms of the binomial series if the parent population be 'infinite,' but of the 

 hypergeometrical series if the parent population be of ' finite ' size. Good approxi- 

 mations to both these series may be obtained from the appropriate Type I curve 

 if the size of the sample be not too small. The following example illustrates the 

 position in the case of the binomial. 



In an indefinitely large population 1/20 of the individuals possess a certain 

 character G. A random sample of 100 is drawn ; and the number of individuals, x, 

 possessing character A is observed. Within what limits may x be "almost certainly" 

 expected to lie ? 



The moment-coefficients of the binomial are as follows : 



1 4o 1 Qpq 



/3i=- *. /3 2 = 3 + - -. 

 npq npq 



It is found that in the present case 



Mean a; = 5-0000, <r x = 21794, ^='1705, /9 2 = 31505. 



Suppose that for the limits within which x is "almost certain" to fall, we take those 

 given by d.oos and ^.995, within which the chances are 99 to 1 that x will fall. The 

 Tables, when entered with the above values of /3i and /8 2 , give on interpolation 



d.oo5 = - 214, do* = + 2-92. 

 The corresponding limits for x are 



tf.oo5 = 5-0000 - 2-14 x 2-1794 = 0'34, 



#.995 = 5-0000 + 2-92 x 2-1794 = 11-36. 

 The true binomial frequencies calculated to four places of decimals are as follows : 



