Fundamentals of Probability and Statistics for Engineers

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
and
following the method of least squares, and let

We see from Equation (11.15) that

where

and Yj,j 1,2,...,n, are independent and identically distributed according to
Equation (11.4). Thus, if we write

70

60

50

40

40 50 60 70 80 90

y

x

y= 17.03 + 0.57x

Figure 11.2 Estimated regression line and observed data for Example 11.1

342 Fundamentals of Probability and Statistics for Engineers

Equation (11.15).Let and denote, respectively, the estimators for
A^ B^

Q^ˆ

A^

B^

"

: … 11 : 16 †

Q^ˆ…CTC†^1 CTY; … 11 : 17 †

Yˆ

Y 1

...

Yn

2

6

4

3

7

5 ; …^11 :^18 †

ˆ

YˆCq‡E; … 11 : 19 †