Page 371 - Fundamentals of Probability and Statistics for Engineers

354 Fundamentals of Probability and Statistics for Engineers
^
0 31 < 0 61, we accept H
Since ˆ : : 0 . That is, we conclude that the data do not
f g
indicate a linear relationship between E Y and x; the probability that we are
wrong in accepting H 0 is 0.05.
In closing, let us remark that we are often called on to perform tests of
simultaneous hypotheses. For example, one may wish to test H 0 : ˆ 0 and
ˆ 1 0o 1 or both. Such tests involve both estimators
0 or 6ˆ
1 against H 1 :: 6ˆ
^
^
A and B and hence require their joint distribution. This is also often the case in
multiple linear regression, to be discussed in the next section. Such tests
customarily involve F-distributed test statistics, and we will not pursue them
here. A general treatment of simultaneous hypotheses testing can be found in
Rao (1965), for example.

11.2 MULTIPLE LINEAR REGRESSION

The vector–matrix approach proposed in the preceding section provides a smooth
transition from simple linear regression to linear regression involving more than
one independent variable. In multiple linear regression, the model takes the form

EfYgˆ 0 ‡ 1 x 1 ‡ 2 x 2 ‡ ‡ m x m : 11:45†

Again, we assume that the variance of Y is 2 and is independent of x 1 , x 2 , .. . , and
x m . As in simple linear regression, we are interested in estimating (m ‡ 1) regres-
sion coefficients 1 ,. . ., and m , obtaining certain interval estimates, and testing
0 ,
hypotheses about these parameters on the basis of a sample of Y values with their
associated values of (x 1 , x 2 , . . ., x m ). Let us note that our sample of size n in this
case takes the form of arrays (x 11 , x 21 , .. ., x m1 , Y 1 ), (x 12 , x 22 , .. . , x m2 , Y 2 ), .. . ,
(x 1n , x 2n , .. ., x mn , Y n ). For each set of values x ki , k ˆ 1, 2, . . . , m, of x i , Y i is an
independent observation from population Y defined by

Y ˆ 0 ‡ 1 x 1 ‡ ‡ m x m ‡ E: 11:46†

As before, E is the random error, with mean 0 and variance 2 .

11.2.1 LEAST SQUARES METHOD OF ESTIMATION

To estimate the regression coefficients, the method of least squares will again be
ˆ
employed. Given observed sample-value sets (x 1i , x 2i ,..., x mi , y i ), i 1, 2,..., n,
the system of observed regression equations in this case takes the form

y i ˆ 0 ‡ 1 x 1i ‡ ‡ m x mi ‡ e i ; i ˆ 1; 2; ... ; n: 11:47†

TLFeBOOK