Introduction to Probability and Statistics for Engineers and Scientists

9.3Distribution of the Estimators 355

Relative humidity

18

16

14

12

10

8

(^625)
Estimated
regression
line
30 35 40 45 50 55 60 65
Moisture content
FIGURE 9.2 Example 9.2a.
These data are plotted in Figure 9.2. To compute the least squares estimator and the
estimated regression line, we run Program 9.2; results are shown in Figure 9.3. ■

9.3Distribution of the Estimators

To specify the distribution of the estimatorsAandB, it is necessary to make additional
assumptions about the random errors aside from just assuming that their mean is 0. The
usual approach is to assume that the random errors are independent normal random
variables having mean 0 and varianceσ^2. That is, we suppose that ifYiis the response
corresponding to the input valuexi, thenYi,...,Ynare independent and

Yi∼N(α+βxi,σ^2 )

Note that the foregoing supposes that the variance of the random error does not depend
on the input value but rather is a constant. This valueσ^2 is not assumed to be known but
rather must be estimated from the data.
Since the least squares estimatorBofβcan be expressed as

B=

∑

i

(xi−x)Yi

∑

i

xi^2 −nx^2

(9.3.1)