table because the coefficient could be either greater or
less than zero. Frequently, a two-tailed table is
required to test the value of interaction terms, for
example. Note that the most commonly available "t" table
is the two-tailed table. Returning to the example, find
that for n k 1 = 6 2 1 = 3 degrees of freedom, the
value of t in a one-tailed table for the 0.025 level of
confidence is 3.182. The value of t for the 0.05 level of
confidence is 2.353 and for the 0.10 level of confidence
the value of t is 1.638. From this, it can be seen that
the coefficient bl is significant at the 2.5% level and b2
is significant at the 10% level. These values of t
(one-tailed) can also be taken from a two-tailed table, as
follows:

 Distribution of t for 3 degrees of freedom
 Probability of a larger value
 One-tailed 0.100 0.050 0.025
 Two-tailed 0.200 0.100 0.050
 t 1.638 2.353 3.182

 The confidence interval for the two bi coefficients
is a function of ta and Sy.12. Obviously, different
values of the bi within the confidence interval will
create different kinds of response surfaces. Errors in
estimating the bi coefficients create wider errors as one
mo"es further from the mean value. Hence, the confidence
band (Fig. VI-7) is narrowest near the mean values of X
and Y. Errors in estimating the coefficients plus error
in estimating the level of the regression create the
confidence band for the regression.

Significance of the Equation

 The t test determines the significance of each of the
coefficients individually. Each coefficient can be
accepted or rejected in accordance with results of the
test. The F test is a test of the significance of the
complete equation and, at the same time, a test of all the
coefficients combined. The value of F is calculated from
already-known values of explained variation and the
variance of the regression, as follows: