CHAPTER 13 Simple Linear Regression

516 CHAPTER THIRTEEN Simple Linear Regression
where
Yˆ
i = predicted value of Y for observation i
X i
= value of X for observation i
b 0
= sample Y intercept
b 1
= sample slope
Equation (13.2) requires the determination of two regression coefficients—b 0
(the sample
Y intercept) and b 1
(the sample slope). The most common approach to finding b 0
and b 1
is the
method of least squares. This method minimizes the sum of the squared differences between
the actual values (Y i
) and the predicted values ( Yˆ i ) using the simple linear regression equation
[that is, the prediction line; see Equation (13.2)]. This sum of squared differences is equal to
n
∑
i=
1
( Y − Yˆ )
i
i
2
Because Yˆ
= b + b X ,
i
0 1
i
n
∑
i=
1
n
2
i i ∑ i 0 1 i
i=
1
( Y − Yˆ ) = [ Y − ( b + b X )]
2
Because this equation has two unknowns, b 0
and b 1
, the sum of squared differences depends on
the sample Y intercept, b 0
, and the sample slope, b 1
. The least-squares method determines the
values of b 0
and b 1
that minimize the sum of squared differences. Any values for b 0
and b 1
other than those determined by the least-squares method result in a greater sum of squared differences
between the actual values (Y i
) and the predicted values Yˆ
i . In this book, Microsoft
Excel is used to perform the computations involved in the least-squares method. For the data of
Table 13.1, Figure 13.4 presents results from Microsoft Excel.
FIGURE 13.4
Microsoft Excel results
for the Sunflowers
Apparel data
See Section E13.1 to create
this.
S YX
n
SSR
SSE
SST
p-value
b 0
b 1