Applied Statistics and Probability for Engineers

440 CHAPTER 12 MULTIPLE LINEAR REGRESSION

the point x 0. This must include the error in estimating the mean at that point, as well as the in-

herent variability in the random variable Yat the same value xx 0.

In predicting new observations and in estimating the mean response at a given point

x 01 ,x 02 ,... , x 0 k, we must be careful about extrapolatingbeyond the region containing the

original observations. It is very possible that a model that fits well in the region of the origi-

nal data will no longer fit well outside of that region. In multiple regression it is often easy to

inadvertently extrapolate, since the levels of the variables (xi 1 , xi 2 ,... , xik), i1, 2,... , n,

jointly define the region containing the data. As an example, consider Fig. 12-5, which illus-

trates the region containing the observations for a two-variable regression model. Note that

the point (x 01 , x 02 ) lies within the ranges of both regressor variables x 1 and x 2 , but it is outside

the region that is actually spanned by the original observations. Thus, either predicting the

value of a new observation or estimating the mean response at this point is an extrapolation of

the original regression model.

EXAMPLE 12-8 Suppose that the engineer in Example 12-1 wishes to construct a 95% prediction interval on

the wire bond pull strength when the wire length is x 1 8 and the die height is x 2 275. Note

that [1 8 275], and the point estimate of the pull strength is

Also, in Example 12-7 we calculated Therefore, from Equation

12-40 we have

and the 95% prediction interval is

Notice that the prediction interval is wider than the confidence interval on the mean response

at the same point, calculated in Example 12-7. The Minitab output in Table 12-4 also displays

this prediction interval.

22.81 Y 0

 32.51

27.662.074 ̨ 2 5.2352 11 0.04444 2

 Y 0

 27.662.074 2 5.2352 11 0.04444 2

x¿ 01 ¿ 2 ^1 x 0 0.04444.

x¿ 0 yˆ 0 x¿ 0 ˆ27.66.

x 01

x 01

x 2

Joint region

of original area

Original range for x 1

x 1

Original range for

x^2

Figure 12-5 An example of extrapolation in

multiple regression.

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