Introduction to Probability and Statistics for Engineers and Scientists

356 Chapter 9: Regression

Simple Linear Regression

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44

Sample size = 15

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49,12

52,14

38,9

55,16

32,8

57,18

54,14

44,12

x =

y = 12

a =

b =

−2.51

0.32

The least squares estimators are as follows:

Average x value =

Sum of squares of the x values =

46.13

33212.0

The estimated regression line is Y = −2.51 + 0.32x

S(x, Y)

S(x, x)

S(Y, Y)

SSR

=

=

=

=

416.2

1287.73

147.6

13.08

100

80

60

40

20

0

-20 50 100 150 200 250 300

FIGURE 9.3

we see that it is a linear combination of the independent normal random variablesYi,
i=1,...,nand so is itself normally distributed. Using Equation 9.3.1, the mean and
variance ofBare computed as follows:

E[B]=

∑

i

(xi−x)E[Yi]

∑

i

xi^2 −nx^2

=

∑

i

(xi−x)(α+βxi)

∑

i

x^2 i−nx^2