Variance, Standard Error,and Confidence Intervals

 The standard error of the estimate of the regression
is the square root of the unexplained variation adjusted
for degrees of freedom. Unexplained variation, or the
variance of the regression, is

 Sy.122 = (Ey2 E9122) / (n k 1)

where k is the number of coefficients estimated (bl and b2
in this example). In the example:

 Sy.122 = (3.2483 2.9879) / (6-2-1)
 = 0.0868

Standard error is the square root of the variance:

 Sy.12 = (Sy.122)1/2
 = (0.0868)1/2
 = 0.29462

This value indicates the precision with which the
regression measures the average value of yield, Y =
6.333 tons/ha. This value is used to calculate the
confidence interval around the mean yield as follows:

 Y  ta [Sy.12 / nl/2]

where ta is the tabulated value of t for the level of
significance a and with n k degrees of freedom.

For example, at the 95% confidence level (a = 0.05) with
n = 6 and with 6 2 = 4 degrees of freedom, the
confidence interval is:

 6.333  (2.776)(0.29462 / 2.45)
or
 6.333  0.334

With this result, there is a 95% confidence that the true
mean yield for the same treatments is included between
5.999 and 6.667 metric tons/ha.