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Part II: Making Predictions by Using Regression
The general format for finding a test statistic in most any situation is to take
the statistic (in this case, the coefficient), subtract the value in Ho (zero), and
divide by the standard error of that statistic (for this example, the standard
error of the coefficient). (For more info on the general format of hypothesis
tests, see Chapter 3.)
To test a regression coefficient, the test statistic (using the labels from Fig-
ure 5-3) is (Coef – 0)/SE Coef. In non-computer language, that means you take
the coefficient, subtract zero, and divided 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. (See 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 prespecified α (usually 0.05), then you reject Ho and
conclude that the coefficient of that x variable isn’t zero and that variable
makes a significant contribution toward estimating y (given the other vari-
ables 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 Table A-1. 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; these results can be found by looking across the news-
paper ads row of Figure 5-3. From this you should include both the TV ads
variable and the newspaper ads variable in the model for estimating TV sales.