CHAPTER 13 Simple Linear Regression
CHAPTER 13 Simple Linear Regression
CHAPTER 13 Simple Linear Regression
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526 <strong>CHAPTER</strong> THIRTEEN <strong>Simple</strong> <strong>Linear</strong> <strong>Regression</strong><br />
The Coefficient of Determination<br />
By themselves, SSR, SSE, and SST provide little information. However, the ratio of the regression<br />
sum of squares (SSR) to the total sum of squares (SST ) measures the proportion of variation<br />
in Y that is explained by the independent variable X in the regression model. This ratio is<br />
called the coefficient of determination, r 2 , and is defined in Equation (<strong>13</strong>.9).<br />
COEFFICIENT OF DETERMINATION<br />
The coefficient of determination is equal to the regression sum of squares (that is,<br />
explained variation) divided by the total sum of squares (that is, total variation).<br />
r<br />
2 <strong>Regression</strong> sum of squares<br />
= =<br />
Total sum of squares<br />
SSR<br />
SST<br />
(<strong>13</strong>.9)<br />
The coefficient of determination measures the proportion of variation in Y that is explained<br />
by the independent variable X in the regression model. For the Sunflowers Apparel data, with<br />
SSR = 105.7476, SSE = 11.2067, and SST = 116.9543,<br />
r 2 105.<br />
7476<br />
= = 0.<br />
9042<br />
116.<br />
9543<br />
Therefore, 90.42% of the variation in annual sales is explained by the variability in the size of the<br />
store, as measured by the square footage. This large r 2 indicates a strong positive linear relationship<br />
between two variables because the use of a regression model has reduced the variability in<br />
predicting annual sales by 90.42%. Only 9.58% of the sample variability in annual sales is due to<br />
factors other than what is accounted for by the linear regression model that uses square footage.<br />
Figure <strong>13</strong>.8 presents the coefficient of determination portion of the Microsoft Excel results<br />
for the Sunflowers Apparel data.<br />
FIGURE <strong>13</strong>.8<br />
Partial Microsoft Excel<br />
regression results for the<br />
Sunflowers Apparel data<br />
S YX<br />
See Section E<strong>13</strong>.1 to create<br />
the worksheet that contains<br />
this area.<br />
EXAMPLE <strong>13</strong>.4<br />
COMPUTING THE COEFFICIENT OF DETERMINATION<br />
Compute the coefficient of determination, r 2 , for the Sunflowers Apparel data.<br />
SOLUTION You can compute SST, SSR, and SSE, that are defined in Equations (<strong>13</strong>.6), (<strong>13</strong>.7),<br />
and (<strong>13</strong>.8) on pages 524–525, by using Equations (<strong>13</strong>.10), (<strong>13</strong>.11), and (<strong>13</strong>.12).<br />
COMPUTATIONAL FORMULA FOR SST<br />
2<br />
⎛ n ⎞<br />
⎜ Y<br />
n<br />
n ∑ i⎟<br />
⎝ i=<br />
⎠<br />
SST = ∑ ( Yi<br />
− Y )2 = ∑Yi<br />
2 −<br />
1 n<br />
i=<br />
1<br />
i=<br />
1<br />
(<strong>13</strong>.10)
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