DISCUSSION OF RESULTS 353 



however, note that the standard errors of estimate for the two hnes are, 

 to two significant figures, equal, indicating that the difference in the 

 slopes is probably statistically insignificant. 



There are some data points in figure 8.26 having a rather large devia- 

 tion from the regression lines. Statistical theory (using the "Student" 

 ^-distribution for 84 deg of freedom) shows that, if the data points are 

 drawn from a normally distributed population, there should be only one 

 point having a deviation of more than ±9 mm from the observed regres- 

 sion line. There are in fact five points in figure 8.26, four above and 

 one below the line. If these five points are thrown out, on the grounds 

 that they weight too heavily the extremes of the distribution of data points 

 (this is especially true when using least squares regression), and the regres- 

 sion is then redone using the remaining 81 data points, the resulting value 

 of the slope is —0.385 mm/A'^-unit with r = 0.77 compared to the pre- 

 dicted slope of —0.381 mm/A'^-unit, a rather close agreement. 



8.3.6. Discussion of Results 



As a summary of the results of the experimental versus theoretical 

 comparisons given in the preceding section, a statistical analysis has been 

 run on the significance of the differences between the slopes of the ob- 

 served and predicted regression lines. In order to make the tests more 

 stringent, it was assumed that the slopes derived from the Standard Sam- 

 ple should be taken to be the slopes of the population regression lines (/3), 

 thus yielding an estimate of the significance of the departure of the ob- 

 served slope from the assumed population value. 



A value of t was first calculated for each case using the relation [25] 



_ I b - ^0 I V2(x. - xY 



where b is the observed slope, ^o the assumed population, or theoretical, 

 slope, X refers to the independent variable in each regression, N s, SE is 

 the standard error of estimate, and ij_2 is the value of t for j — 2 deg of 

 freedom. From tj-2, confidence limits for /3 at the 100(1— a) percent 

 level can be calculated from [25] 



lj^,aSE < ^ < ^ ^ tj^2,.SE ^g_26) 



VS(Xi - x)2 \/2(Xi - 



The probability that the observed value b would have fallen outside of 

 these limits by chance is a. Many statisticians consider a value of tj^o 



