CHAPTER 13 Simple Linear Regression

528 CHAPTER THIRTEEN Simple Linear Regression
Standard Error of the Estimate
Although the least-squares method results in the line that fits the data with the minimum
amount of error, unless all the observed data points fall on a straight line, the prediction line is
not a perfect predictor. Just as all data values cannot be expected to be exactly equal to their
mean, neither can they be expected to fall exactly on the prediction line. An important statistic,
called the standard error of the estimate, measures the variability of the actual Y values from
the predicted Y values in the same way that the standard deviation in Chapter 3 measures the
variability of each value around the sample mean. In other words, the standard error of the estimate
is the standard deviation around the prediction line, whereas the standard deviation in
Chapter 3 is the standard deviation around the sample mean.
Figure 13.5 on page 517 illustrates the variability around the prediction line for the
Sunflowers Apparel data. Observe that although many of the actual values of Y fall near the
prediction line, none of the values are exactly on the line.
The standard error of the estimate, represented by the symbol S YX
, is defined in Equation
(13.13).
STANDARD ERROR OF THE ESTIMATE
S
YX
=
( Yi
− Yˆ i)
SSE i=
1
=
n − 2 n − 2
n
∑
2
(13.13)
where
Y i
= actual value of Y for a given X i
ˆ Y i
= predicted value of Y for a given X i
SSE = error sum of squares
From Equation (13.8) and Figure 13.4 on page 516, SSE = 11.2067. Thus,
S YX =
11.
2067
14 − 2
= 0.
9664
This standard error of the estimate, equal to 0.9664 millions of dollars (that is, $966,400), is
labeled Standard Error in the Microsoft Excel results shown in Figure 13.8 on page 526. The standard
error of the estimate represents a measure of the variation around the prediction line. It is
measured in the same units as the dependent variable Y. The interpretation of the standard error of
the estimate is similar to that of the standard deviation. Just as the standard deviation measures
variability around the mean, the standard error of the estimate measures variability around the
prediction line. For Sunflowers Apparel, the typical difference between actual annual sales at a
store and the predicted annual sales using the regression equation is approximately $966,400.
PROBLEMS FOR SECTION 13.3
Learning the Basics
PH Grade
ASSIST
PH Grade
ASSIST
13.11 How do you interpret a coefficient of
determination, r 2 , equal to 0.80?
13.12 If SSR = 36 and SSE = 4, determine SST
and then compute the coefficient of determination,
r 2 , and interpret its meaning.
PH Grade
ASSIST
PH Grade
ASSIST
13.13 If SSR = 66 and SST = 88, compute the
coefficient of determination, r 2 , and interpret its
meaning.
13.14 If SSE = 10 and SSR = 30, compute the
coefficient of determination, r 2 , and interpret its
meaning.