Applied Statistics and Probability for Engineers

The fittedor estimated regression lineis therefore

(11-9)

Note that each pair of observations satisfies the relationship

where eiyi is called the residual.The residual describes the error in the fit of the

model to the ith observation yi. Later in this chapter we will use the residuals to provide in-

formation about the adequacy of the fitted model.

Notationally, it is occasionally convenient to give special symbols to the numerator and

denominator of Equation 11-8. Given data (x 1 , y 1 ), (x 2 , y 2 ), p, (xn, yn), let

(11-10)

and

(11-11)

EXAMPLE 11-1 We will fit a simple linear regression model to the oxygen purity data in Table 11-1. The

following quantities may be computed:

(^) a
20
i 1
yi^2 170,044.5321 (^) a
20
i 1
xi^2 29.2892 (^) a
20
i 1
xiyi2,214.6566
n (^20) a
20
i 1
xi23.92 (^) a
20
i 1

yi1,843.21 x1.1960 y92.1605

Sx y a

n

i 1

yi 1 xi x 22  a

n

i 1

xiyi

aa

n

i 1

xib

aa

n

i 1

yib

n

Sx x a

n

i 1

1 xi x 22  a

n

i 1

x (^2) i
aa
n
i 1
xib
2
n
yˆi

yiˆ 0 ˆ 1 xiei, i1, 2,p, n

yˆˆ 0 ˆ 1 x

The least squares estimatesof the intercept and slope in the simple linear regression

model are

(11-7)

(11-8)

where y 11

 n 2 g

n

i 1 yi and x^11 n^2 g

n

i 1 xi.

ˆ 1 

a

n

i 1

yi xi

aa

n

i 1

yib aa

n

i 1

xib

n

a

n

i 1

x (^2) i
aa
n
i 1
xib
2
n
ˆ 0 y ˆ 1 x
Definition
11-2 SIMPLE LINEAR REGRESSION 377
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