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Chapter 5: Multiple Regression with Two X Variables 95

You can more easily interpret the number “0.16211 million dollars” by convert-
ing 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 increase 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 increase by
0.24887 million dollars (or $248,870) when newspaper 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 Stats II. If you were to forget about units
in the ads and sales example, you would think that sales increased by 0.24887
dollars with $1 in newspaper ad spending!
Knowing the multiple regression coefficients (b and b , in this case) and their
1 2
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 also can say how much you expect sales to increase per $1,000
you spend on TV or newspaper advertising. Note that this conclusion 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. The next
section tells you what to do next.

Testing the coefficients

To officially determine whether you have the right x variables in your mul-
tiple regression model, do a formal hypothesis test to make sure the coef-
ficients aren’t equal to zero. Note that if the coefficient of an x variable is
zero, 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 variable’s
coefficient is equal to zero, you don’t need that x variable in the model.

With any regression analysis, the computer automatically performs all the
necessary hypothesis tests for the regression coefficients. Along with the
regression 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.

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.)

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