Applied Statistics and Probability for Engineers

factorial points, and let be the average of the nCrun at the center point. If the difference

is small, the center points lie on or near the plane passing through the factorial points,

and there is no curvature. On the other hand, if is large, curvature is present. A single-

degree-of-freedom sum of squares for curvature is given by

(S14-12)

where, in general, nFis the number of factorial design points. This quantity may be compared

to the error mean square to test for curvature. Notice that when Equation S14-12 is divided by

, the result is similar to the square of the tstatistic used to compare two means.

More specifically, when points are added to the center of the 2kdesign, the model we may

entertain is

where the jjare pure quadratic effects. The test for curvature actually tests the hypotheses

Furthermore, if the factorial points in the design are unreplicated, we may use the nCcenter

points to construct an estimate of error with nC 1 degrees of freedom.

EXAMPLE S14-2 A chemical engineer is studying the percentage of conversion or yield of a process. There are

two variables of interest, reaction time and reaction temperature. Because she is uncertain about

the assumption of linearity over the region of exploration, the engineer decides to conduct a 2^2

H 1 : a

k

j 1

jj^0

H 0 : a

k

j 1

jj^0

Y 0 a

k

j 1

jxjb

ij

ijxixja

k

j 1

jjx^2 j

ˆ^2 MSE



°

yF yC

B

1

nF

1

nC

¢

2

SSCurvature

nFnC 1 yF yC 22

nFnC

yF yC

yF yC

yC

14-10

• 1+1 0

35

A = Reaction time (min)

30 40

0

• 1

1

155

150

160

39.3 40.9

40.3

40.5

40.7

40.2

40.6

40.0 41.5

B = Temperature (

°C)

Figure S14-4 The 2^2

design with five center

points for Example

S14-2.

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