Sec. 7.2 DETERMINING LINEAR TREND LINES 203 



results shown specifically in Table 7.22 and graphically as X's in 

 Figure 7.21. The decision regarding linearity of trend and the esti- 

 mation of any desired features of the true trend line (such as slope) 

 must be accomplished from the information contained in the sample. 

 If the trend of one variable with another is linear, the relationship 

 between the two kinds of measurements, X and Y, is of the form 

 Y = A + BX, in which, for illustration, Y stands for the 28-week 

 weight of a certain breed of turkey and X = the 16-week weight of 



TABLE 7.22 



A Random Sample of n = 30 Pairs (X, Y) prom Table 7.21 



X Y X Y X Y X Y 



the same turkey. If all the observed pairs of measurements (X, Y) 

 in Table 7.21 satisfy a linear equation perfectly, all the points of 

 Figure 7.21 will lie exactly on the same straight line; and the rela- 

 tionship between the two variables will be perfectly linear. More- 

 over, the equation of the line can be determined from the coordinates 

 of any two distinct points. Such obviously is not the case in Figure 

 7.21 because errors of measurement and uncontrollable fluctuations 

 in the 28-week weights of turkeys which weighed the same at 16 

 weeks of age must be averaged out before the trend appears to be 

 linear. By contrast with situations met in elementary algebra, where 

 the equation is given and all appropriate points lie on the line, the 

 present situation starts with the points given from sample observa- 

 tions, and the problem is to determine which straight line best fits 

 these observations, and, it is hoped, best estimates the true linear 

 regression line. 



Assuming that a set of observations really does follow a linear 

 trend quite well, how can a specific equation of the form Y = A + BX 

 be determined and also defended as the best straight line to be em- 

 ployed under the circumstances? The answer to this question de- 

 pends upon the interpretation of the word "best." One interpreta- 

 tion, and the one most frequently accepted, can be illustrated by 



