224 LINEAR REGRESSION AND CORRELATION Ch. 7 



come from sampling studies, but perhaps they occur in practice often 

 enough to justify the inclusion here of a method for obtaining the 

 r and the b. 



As in Chapter 2, the computations will be carried out in units of 

 the class intervals. A two-way frequency distribution table is needed 

 because two variables are involved. These matters, and others, are 

 illustrated and discussed by means of 280 pairs of observations of 

 16- and 28- week weights of female turkeys similar to those studied 

 earlier in this chapter. The symbol, X, is used to denote the 16-week 

 weights and Y will stand for the 28-week weights, as before. Now 

 that two variables are being considered simultaneously, frequencies 

 in the X-classes will be symbolized by f x , those for the 7-classes by 

 f Y . When it is desirable to indicate both the X and the Y for a class 

 of data, j Y x will denote the frequency in that "cell" in the two-way 

 table. Also, there may be two different lengths of class interval, I x 

 and I Y for X and Y, respectively. With these symbols in mind, the 

 following formulas are seen to be analogous to those used previously 

 for b and r: 



, 2(fxvdx-d Y ) - [®fz-dz)®f7-d T )]m . 



b = n and 



SQWx 2 ) - (2/W A 2 /2/x 



same numerator as that above for b 



f sz — ■ — — • 



V (same as denominator above) (same with Y replacing X) 



The data of Table 7.42 are arranged in a two-way frequency dis- 

 tribution table to provide a relatively easy basis for calculating b 

 and r from their formulas as given above. 



The following computations are derived from the summaries in 

 Table 7.42: 



Zifr-dr 2 ) - (S/ydF) 2 /2/ F = 800.5714, and its square root = 28.65. 



2(Jx-d x 2 ) ~ (2fx-d x ) 2 /2fx = 924.5679, and its square root = 30.41. 



2(fxY-d x -d Y ) - [(S/z-dj)(S/yd r )]/S/= 448.2857. 



PROBLEMS 



1. Calculate the r for the data of problem 1, section 7.1. 



2. Calculate as in problem 1 for the data of problem 2, section 7.1. Given 

 SA' 2 = 18,484, SAF = 727.99, SF 2 = 28.6918. Ans. r = +.96. 



3. Compute 2(F — F) 2 for the data of problem 3, section 7.1 by using the for- 

 mula: 2(F - F) 2 - (1 - r 2 )-2(y 2 ). 



