Fundamentals of Probability and Statistics for Engineers

Answer: in this case, C is a 12 3 matrix and

We thus have, upon finding the inverse of CTC by using either matrix inversion
formulae or readily available matrix inversion computer programs,

or

The estimated regression equation based on the data is thus

Since Equation (11.48) is identical to its counterpart in the case of simple linear
regression, much of the results obtained therein concerning properties of least-
square estimators, confidence intervals, and hypotheses testing can be dupli-
cated here with, of course, due regard to the new definitions for matrix C and
vector.
Let us write estimator for in the form

We see immediately that

Hence, least-square estimator is again unbiased. It also follows from Equa-

356 Fundamentals of Probability and Statistics for Engineers

tion (11.21)that the covariance matrix for isgivenby

CTCˆ

12 626 290

626 36;776 15; 336

290 15;336 7; 028

2

6

4

3

7

5 ;

CTyˆ

2 ; 974

159 ; 011

72 ; 166

2

6

4

3

7

5 :

^qˆ…CTC†^1 CTyˆ

 33 : 84

0 : 39

10 : 80

;

^ 0 ˆ 33 : 84 ; ^ 1 ˆ 0 : 39 ; ^ 3 ˆ 10 : 80 :

Edfygˆ ^ 0 ‡ ^ 1 x 1 ‡ ^ 2 x 2

ˆ 33 : 84 ‡ 0 : 39 x 1 ‡ 10 : 80 x 2 :

q

Q^ q

Q^ˆ…CTC†^1 CTY: … 11 : 50 †

EfQ^gˆ…CTC†^1 CTEfYgˆq: … 11 : 51 †

Q^

Q^

covfQ^gˆ^2 …CTC†^1 : … 11 : 52 †

2

4

3

5

;