Introduction to Probability and Statistics for Engineers and Scientists

9.9Polynomial Regression 393

might hold. Since

∑

i

xi=55,

∑

i

xi^2 =385,

∑

i

xi^3 =3,025,

∑

i

xi^4 =25, 333

∑

i

Yi=1,291.1,

∑

i

xiYi=9,549.3,

∑

i

xi^2 Yi=77,758.9

the least squares estimates are the solution of the following set of equations.

1,291.1= 10 B 0 + 55 B 1 + 385 B 2 (9.9.1)

9,549.3= 55 B 0 + 385 B 1 +3,025B 2

77,758.9= 385 B 0 +3,025B 1 +25,333B 2

Solving these equations (see the remark following this example) yields that the least
squares estimates are

B 0 =12.59326, B 1 =6.326172, B 2 =2.122818

Thus, the estimated quadratic regression equation is

Y=12.59+6.33x+2.12x^2

This equation, along with the data, is plotted in Figure 9.14. ■

300

250

200

150

100

50

0

Y

246810

x

FIGURE 9.14