Fundamentals of Probability and Statistics for Engineers

Since 0. That is, we conclude that the data do not
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 :0and
1 against H 1 : 0 or 1 or both. Such tests involve both estimators

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
R ao (1965), for example.

11.2 M ultiple Linear R egression

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

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

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

11.2.1 Least Squares M ethod of Estimation

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

354 Fundamentals of Probability and Statistics for Engineers

0 31 < 0 61, we accept H

and andhence requiretheir jointdistribution.This is also often thecase in

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: :

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