Introduction to Probability and Statistics for Engineers and Scientists

9.7Transforming to Linearity 381

9.7Transforming to Linearity

In many situations, it is clear that the mean response is not a linear function of the input
level. In such cases, if the form of the relationship can be determined it is sometimes
possible, by a change of variables, to transform it into a linear form. For instance, in
certain applications it is known thatW(t), the amplitude of a signal a timetafter its
origination, is approximately related totby the functional form

W(t)≈ce−dt

On taking logarithms, this can be expressed as

logW(t)≈logc−dt

If we now let

Y=logW(t)

α=logc

β=−d

then the foregoing can be modeled as a regression of the form

Y=α+βt+e

The regression parametersαandβwould then be estimated by the usual least squares
approach and the original functional relationships can be predicted from

W(t)≈eA+Bt

EXAMPLE 9.7a The following table gives the percentages of a chemical that were used up
when an experiment was run at various temperatures (in degrees celsius). Use it to estimate
the percentage of the chemical that would be used up if the experiment were to be run at
350 degrees.

Temperature Percentage

5 ◦ .061

10 ◦ .113

20 ◦ .192

30 ◦ .259

40 ◦ .339

50 ◦ .401

60 ◦ .461

80 ◦ .551