Significance of the Coefficients


 The confidence interval about the mean is a way to
measure the error associated with the level of the
response surface. It is also possible to have errors in
measuring the slope and the curvature of the response
surface, represented by the values of bl and b2 in the
example. The usual test is to determine whether the
values of the b's (the regression coefficients)
individually are different from zero. This is done by the
use of the "t" test. If the value of t is not sufficient
to provide confidence that the value of a certain bi is
different from zero (either positively or negatively) it
is concluded that yield is not affected by the term
corresponding to the bi being tested. This follows,
because if the value of bi is not different from zero,
then it must equal zero and the term disappears from the
equation. Hence, in the equation

 Y = 5.350 + 1.225 N 0.228 N2

if it is determined that b2 (-0.228) is not different from
zero, the N2 will disappear because it is multiplied by
zero and the equation left is of the form Y = a + bN.
The value of b in the new equation would have to be
calculated again because it would be different from bl in
the quadratic form.
 The t values for each of the coefficients are
calculated as follows:

 tl = bl / [(Sy.12) (x22 / D)1/2]
 = 1.225 / [(0.29462)(87.75 / 60.75)1/2]
 = 3.459

 t2 = b2 / [(Sy.12)(EX22 / D)1/2]
 = -0.228 / [(0.29462)(9.0 / 60.75)1/2]
 = -2.0160

 Because it is known that the value of bl should be
positive and the value of b2 should be negative, it is
possible to use a one-tailed "t" table for testing the
level of significance of these coefficients. If one does
not know beforehand whether a coefficient should be
positive or negative, it is necessary to use a two-tailed