Page 112 - Statistics II for Dummies

96 Part II: Using Different Types of Regression to Make Predictions

To test a regression coefficient, the test statistic (using the labels from
Figure 5-3) is (Coef – 0)/SE Coef. In noncomputer language, that means you
take the coefficient, subtract zero, and divide by the standard error (SE) of
the coefficient. The standard error of a coefficient here is a measure of how
much the coefficient is expected to vary when you take a new sample. (Refer
to Chapter 3 for more on standard error.)

The test statistic has a t-distribution with n – k – 1 degrees of freedom, where
n equals the sample size and k is the number of predictors (x variables) in
the model. This number of degrees of freedom works for any coefficient in
the model (except you don’t bother with a test for the constant, because it
has no x variable associated with it).

The test statistic for testing each coefficient is listed in the column marked
T (because it has a t-distribution) on the Minitab output. You compare the
value of the test statistic to the t-distribution with n – k – 1 degrees of free-
dom (using Table A-1 in the appendix) and come up with your p-value. If the
p-value is less than your predetermined α (usually 0.05), then you reject Ho
and conclude that the coefficient of that x variable isn’t zero and that vari-
able makes a significant contribution toward estimating y (given the other
variables are also included in the model). If the p-value is larger than 0.05,
you can’t reject Ho, so that x variable makes no significant contribution
toward estimating y (when the other variables are included in the model).

In the case of the ads and plasma TV sales example, Figure 5-3 shows that
the coefficient for the TV ads is 0.1621 (the second number in column two).
The standard error is listed as being 0.0132 (the second number in column
three). To find the test statistic for TV ads, take 0.1621 minus zero and divide
by the standard error, 0.0132. You get a value of t = 12.29, which is the second
number in column four. Comparing this value of t to a t-distribution with
n – k – 1 = 22 – 2 – 1 = 19 degrees of freedom (Table A-1 in the appendix),
you see the value of t is way off the scale. That means the p-value is smaller
than can be measured on the t-table. Minitab lists the p-value in column five
of Figure 5-3 as 0.000 (meaning it’s less than 0.001). This result leads you to
conclude that the coefficient for TV ads is statistically significant, and TV ads
should be included in the model for predicting TV sales.

The newspaper ads coefficient is also significant with a p-value of 0.000 by
the same reasoning; you find these results by looking across the newspaper
ads row of Figure 5-3. Based on your coefficient tests and the lack of multico-
linearity between TV and newspaper ads (see the earlier section “Checking
for Multiconlinearity”), you should include both the TV ads variable and the
newspaper ads variable in the model for estimating TV sales.

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