Page 377 - Fundamentals of Probability and Statistics for Engineers

360 Fundamentals of Probability and Statistics for Engineers

where w i are assigned weights. In vector–matrix notation, show that estimates
^ and ^ now take the form
^

^ ˆ C WC† C Wy;
1
T
T
q ˆ
^
where
2 3
w 1 0
w 2
6 7
. 7:
6 7
.
W ˆ 6
4 . 5
0 w n
11.5 (a) In simple linear regression [Equation (11.4)]. use vector–matrix notation and
show that the unbiased estimator for 2 given by Equation (11.33) can be
written in the form
1
^ T
^
c 2
ˆ ‰ Y CQ† Y CQ†Š:
n 2
(b) In multiple linear regression [Equation (11.46)], show that an unbiased esti-
mator for 2 is given by
1
^
^ T
c 2
ˆ ‰ Y CQ† Y CQ†Š:
n m 1
11.6 Given the data in Table 11.6:
Table 11.6 Data for Problem 11.6
x 0 1 2 3 4 5 6 7 8 9
y 3.2 3.1 3.9 4.7 4.3 4.4 4.8 5.3 5.9 6.0
(a) Determine the least-square estimates of and in the linear regression
equation

Y ˆ ‡ x ‡ E:
(b) Determine an unbiased estimate of 2 , the variance of Y .
(c) Estimate E Y at x ˆ 5.
f g
(d) Determine a 95% confidence interval for .
(e) Determine a 95% confidence band for ‡ x.
11.7 In transportation studies, it is assumed that, on average, peak vehicle noise level
(Y ) is linearly related to the logarithm of vehicle speed (v). Some measurements
taken for a class of light vehicles are given in Table 11.7. Assuming that

Y ˆ ‡ log v ‡ E;
10

TLFeBOOK