Sec. 7.3 STANDARD DEVIATION ABOUT TREND LINE 211 



by the standard deviation) has been reduced 100(0.48)/1.58 = 30.4 

 per cent by taking the linear relation between the two measurements 

 into account statistically. Such success in accounting for part of the 

 variation among the measurements, Y t , clearly is important in statisti- 

 cal analyses because the only occasion for such analyses arises as a 

 result of variability among numerical measurements. 



The standard deviation about the trend line, s y . x , also is specifically 

 useful in certain applications of linear trend analysis, two of which will 

 be considered. The regression coefficient, b, estimates the average 

 change in the F-measurement for each unit increase in the X-measure- 

 ment. Its accuracy as such a measure is of interest, and its accuracy 

 is measured by its standard deviation. The standard deviation of b 

 is shown in more advanced statistics courses to be 



(7.34) s b 



For the data of Table 7.22: S(F - F) 2 = 12.9226, 2x 2 = 37.9120, 

 n = 30, and hen ce s y . x = V12.9226/28 = 0.679. Therefore, s b = 

 0.679/V37.9120 = 0.110, approximately. 

 It can be shown that the ratio, 



(7.35) t = (6 - 0)/s b , 



where (3 = the true regression coefficient which is estimated by b, 

 follows the same ^-distribution as that summarized in Table IV with 

 n — 2 degrees of freedom. Therefore, a confidence interval can be 

 computed for f3, and it can be interpreted in the manner previously 

 shown. For Table 7.22, the 95 per cent confidence interval is ob- 

 tained as follows: 



-2.05 < (0.6072 - 0)/O.llO < +2.05 



will be a true inequality for 95 per cent of all samples with 28 de- 

 grees of freedom. Hence the 95 per cent confidence interval is found 

 to be as follows after some simplification of the preceding inequality: 



(7.36) 0.38 < p < 0.83. 



It would be concluded in practice that the slope of the true linear 

 regression line is some value between 0.38 and 0.83, but it is recog- 

 nized that there are 5 chances in 100 that the sample has led us to a 

 false statement. A more useful statement might be that it is esti- 

 mated from (7.36) that a turkey which is one pound heavier than 

 another at 16 weeks of age will, on the average, be 0.38 to 0.83 pound 



