212 LINEAR REGRESSION AND CORRELATION Ch. 7 



heavier at 28 weeks of age. That is, the lighter turkeys at 16 weeks 

 tend to catch up some, but they usually remain 0.38 to 0.83 pound 

 lighter at 28 weeks for each pound that they were lighter at 16 weeks 

 of age. 



Another application of linear trend analysis which makes use of 

 s y . x is one in which Y is to be estimated for some unobserved value of 

 X; for instance, for X = 9.5 pounds at 16 weeks. If X is set equal to 

 9.5 in formula 7.24, Y = 0.6072(9.5) + 10.04 = 15.8 pounds at 28 

 weeks of age. How reliable is this estimate? A look at the scatter 

 diagram leaves only the impression that this estimate should be fairly 

 reliable; hence a more specific measure of its accuracy is needed. The 

 standard deviation of Y is given by the following formula: 



(7.37) st = s y . x Vl/n + (X - x) 2 /Xx 2 , 



where X is the value used to calculate the Y. This estimate of the 

 standard deviation of Y is based on n — 2 degrees of freedom, as ex- 

 plained earlier. It will be convenient in subsequent discussions to add 

 "with n — 2 D/F" after an estimate of this sort to indicate the num- 

 ber of chance deviations upon which the estimate is based. In the 

 example considered in this paragraph, 



s Y = 0.679A/1/30 + (9.5 - 7.56) 2 /37.912 = 0.679(.364) 



= 0.247, with 28 D/F. 



This standard deviation applies when the X's have been chosen in 

 advance and are not subject to sampling error. As noted earlier, the 

 b is then an unbiased sample estimate of the population parameter, /3. 

 Under these circumstances the formula 7.37 can be partially ration- 

 alized in lieu of a more rigorous demonstration of its validity. The Y 

 for a particular X, say X{, is obtained from F = y + (X{ — x)b = y 

 + X{b. Hence the variance of Yi is obtained from the variance of a 

 sum, y -f- X{b, in which the Xi is a fixed number. The variance of y for 

 this particular X will be one-nth of the variance about the trend line, 

 or Sy. x 2 /n. The variance of 6 is Sy^/Hx 2 , as noted earlier, and X{ is a 

 constant; hence the variance of Xib = Xi 2 -s y . x 2 /2(x ). If the variance 

 of the sum, y -f- x^, is just the sum of the variances of those two terms, 

 it follows that 



n z,(x ) 



so that S? = s ri Vl/n + x'i 2 /2(x 2 ) , as in formula 7.37 for a particu- 



