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Chapter 5: When Two Variables Are Better than One: Multiple Regression
In the ads and sales regression analysis (see Figure 5-3), the coefficient of x 1
(TV ad spending) equals 0.16211. So y (plasma TV sales) increases by 0.16211
million dollars when TV ad spending increases by $1,000 and spending on
newspaper ads doesn’t change. (Note that keeping more digits after the deci-
mal point reduces rounding error when in units of millions.)
You can more easily interpret the number 0.16211 million dollars by converting
it to a dollar amount without the decimal point: $0.16211 million is equal to
$162,110. (To get this value, I just multiplied 0.16211 by 1,000,000.) So plasma
TV sales increases by $162,110 for each $1,000 increase in TV ad spending and
newspaper ad spending remains the same.
Similarly, the coefficient of x 2 (newspaper ad spending) equals 0.24887. So
plasma TV sales increases by 0.24887 million dollars (or $248,870) when news-
paper ad spending increases by $1,000 and TV ad spending remains the same.
Don’t forget the units of each variable in a multiple regression analysis.
This mistake is one of the most common in intermediate statistics. If you 99
forgot about units in the ads and sales example, you would think that sales
increased by 0.24887 dollars with a dollar in newspaper ad spending!
Knowing the multiple regression coefficients (b 1 and b 2 , in this case) and
their interpretation, you can now answer the original question: Is the money
spent on TV or newspaper ads worth it? The answer is a resounding Yes!
Not only that, but you can also say how much you expect sales to increase
per $1,000 you spend on TV or newspaper advertising. Note that this conclu-
sion assumes the model fits the data well. You have some evidence of that
through the scatterplots and correlation tests, but more checking needs to
be done before you can run to your manager and tell her the good news. (See
the section “Testing the coefficients” to figure out what to do next.)
Testing the coefficients
Another step in determining whether you have the right x variables in your
multiple regression model is to do a formal hypothesis test to make sure the
coefficients are not equal to zero. Note that if the coefficient of an x variable is
zero, then when you put that coefficient into the model, you get zero times that
x variable (which equals zero). This result is essentially saying that if an x vari-
able’s coefficient is equal to zero, you don’t need that x variable in the model.
The computer performs all the necessary hypothesis tests for the regression
coefficients automatically with any regression analysis. Along with the regres-
sion coefficients you can find on the computer output, you see the test
statistics and p-values for a test of each of those coefficients in the same
row for each coefficient. Each one is testing Ho: Coefficient = 0 versus Ha:
Coefficient ≠ 0.