Applied Statistics and Probability for Engineers

12-4 PREDICTION OF NEW OBSERVATIONS 439

The estimated mean response at this point is found from Equation 12-35 as

The variance of is estimated by

Therefore, a 95% CI on the mean pull strength at this point is found from Equation 12-38 as

which reduces to

Some computer software packages will provide estimates of the mean for a point of interest x 0

and the associated CI. Table 12-4 shows the Minitab output for Example 12-7. Both the esti-

mate of the mean and the 95% CI are provided.

12-4 PREDICTION OF NEW OBSERVATIONS

A regression model can be used to predict new or future observationson the response

variable Ycorresponding to particular values of the independent variables, say, x 01 , x 02 , p, x 0 k.

If , a point estimate of the future observation Y 0 at the point x 01 ,

x 02 , p, x 0 kis

(12-39)

A 100(1)% prediction intervalfor this future observationis

yˆ 0 x¿ 0 ̨ˆ

x¿ 0  3 1, x 01 , x 02 ,p, x 0 k 4

26.66

Y|x 0

 28.66

27.662.074 1 0.23266

Y 0 x 0

 27.662.074 1 0.23266

5.2352 1 0.04444 2 0.23266

£

.214653 .007491 .000340

.007491 .001671 .000019

.000340 .000019 .0000015

§ £

1

8

275

ˆ^2 x 0 ¿ 1 ¿ 2 ^1 x 0 5.2352 3 1 8 275 4 §

ˆY 0 x 0

ˆY|x 0 x 0 ¿ˆ 31 8 2754 £

2.26379

2.74427

0.01253

§27.66

This prediction interval is a generalization of the prediction interval given in Equation 11-33

for a future observation in simple linear regression. If you compare the prediction interval

Equation 12-40 with the expression for the confidence interval on the mean, Equation 12-38,

you will observe that the prediction interval is always wider than the confidence interval. The

confidence interval expresses the error in estimating the mean of a distribution, while the pre-

diction interval expresses the error in predicting a future observation from the distribution at

Y 0

 yˆ 0 t2,np 2 ˆ^2 ̨ 11 x¿ 01 ¿ 2 ^1 x 02 (12-40)

yˆ 0 t2,np 2 ˆ^2 ̨ 11 x 0 ¿ 1 ¿ 2 ^1 x 02

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