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342 Fundamentals of Probability and Statistics for Engineers

y

70
y = 17.03 + 0.57x
60

50

40 x
40 50 60 70 80 90
Figure 11.2 Estimated regression line and observed data for Example 11.1

linear regression produces meaningless results even if a straight line appears
to provide a good fit to the data.

11.1.2 PROPERTIES OF LEAST-SQUARE ESTIMATORS

The properties of the estimators for regression coefficients and can be
determined in a straightforward fashion following the vector–matrix expression
^
^
Equation (11.15). Let A and B denote, respectively, the estimators for and
following the method of least squares, and let
"#
^
^
Q ˆ A : 11:16†
^
B
We see from Equation (11.15) that

^
T
1
T
Q ˆ C C† C Y; 11:17†
where
2 3
Y 1
. 5; 11:18†
6 . 7
.
Y ˆ 4
Y n
and Y j , j ˆ 1, 2, .. . , n, are independent and identically distributed according to
Equation (11.4). Thus, if we write

Y ˆ Cq ‡ E; 11:19†

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