cl Tables for Statisticians and Biometricians [XXVII XXX 



8. Distribution of the Regression Coefficient. 



We now turn to the distribution of the regression coefficient RI = roi/o- 2 and 

 the standard deviation of arrays <r ai = <TI Vl r 2 in small samples. 



We need formulae for their means and their standard deviations, i.e. RI, a Rl , a ai 

 and G 0a . Their values are as follows: 



Ri = pS*fc ................................. (xxi), 



i.e. the value in the parent population, 



^=7= ^l^V ........................ (xxii); 



vn 3 1 2 

 the approximate value of a Rl for large samples has long been known, equalling 



The frequency curve for the distribution of regression coefficients, where 



No,**- 1 ran) 



is *=-F= - --- r ......... (xxin). 



VTT T (i(n - 1)) 



Thus the slopes of the regression lines in samples vary symmetrically round the 

 parent slope p2i/2 2 , but the distribution is given by a curve of Type I\ T and not 

 by a normal curve. For this curve 



which measure the approach to normality as n increases. 



The probability of RI lying within any range x from the mean value Ri 

 i.e. x= RI pSi/2 2 , can be found from 



l^ft.Kn-l)) 

 2 (i,i(n-l))' 



where u = a?/(a 2 + a?) and 5 M (|, |(?i 1)) and -B(^,^(?i 1)) are the incomplete 

 and complete B-functions. 



The correlation surface of RI and <r 2 is given by 



- 2 -lS( 1 ^ ! ?**S J ^W 11 - 1 



e * 2 v J J / ( ...(xxv). 



A /o_ oi In 3) p/i /, i \\ 

 V ZTT i a x ' 1 (^ ^w. 1)^ ^ A 



Clearly, for a given value of a 2 , the distribution curve of J2i is a normal curve 

 of mean =p S -= p2i/2 2 and standard deviation 2 X Vl - p 2 /(\/n o- 2 ). Thus the regres- 

 sion line of RI on <r 2 is horizontal, or the coefficient of correlation r RllT2 is zero. This 



