218 LINEAR REGRESSION AND CORRELATION Ch. 7 



by the Greek letter p (rho), which succinctly describes the degree of 

 scatter of the population points about the true linear trend line as, 

 for example, in Figure 7.21. If p = 1, all the points will lie on the 

 regression line. Since they do not — in Figure 7.21 — there is sampling 

 error in the estimation of p, and hence the r varies from sample to 

 sample. This is similar to the situation when the true regression 

 coefficient, /?, was being estimated from samples. 



If the p is zero, all the sampling estimates r t will not be zero but 

 will have a sampling distribution which is nearly normal in form. 

 In such circumstances it can be shown that the following ratio follows 

 the ^-distribution with n — 2 D/F. Thus 



r rvn — 2 



can be used in the usual manner to test the hypothesis H {p = 0). 

 As was seen in earlier discussions, H will be rejected whenever the 

 size of t becomes so great that it is unreasonable — according to some 

 predetermined standard — to believe that this t is the product of sam- 

 pling variation. 



It is more difficult to place a confidence interval on p than on /? 

 because r is not nearly normally distributed when p^O. This 

 process of computing a confidence interval on p will be discussed and 

 developed somewhat heuristically by means of the empirical data 

 found in Table 7.41. These data were obtained by drawing random 



TABLE 7.41 



Observed Sampling Distributions of r and z = 1/2 LoG e [(1 + r)/(l — r)] 

 for n = 12 and p = +.749 



