Introduction to Probability and Statistics for Engineers and Scientists

382 Chapter 9: Regression

x

P

0.6

0.5

0.4

0.3

0.2

0.1

0.0

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FIGURE 9.10 Example 9.7a.

SOLUTION LetP(x) be the percentage of the chemical that is used up when the experiment
is run at 10xdegrees. Even though a plot ofP(x) looks roughly linear (see Figure 9.10),
we can improve upon the fit by considering a nonlinear relationship betweenxandP(x).
Specifically, let us consider a relationship of the form

1 −P(x)≈c(1−d)x

That is, let us suppose that the percentage of the chemical that survives an experiment run
at temperaturexapproximately decreases at an exponential rate whenxincreases. Taking
logs, the preceding can be written as

log(1−P(x))≈log(c)+xlog(1−d)

Thus, setting

Y=−log(1−P)

α=−logc

β=−log(1−d)

we obtain the usual regression equation

Y=α+βx+e