Fundamentals of Probability and Statistics for Engineers

where wi are assigned weights. In vector–matrix notation, show that estimates

and now take the form

where

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

(b) In multiple linear regression [Equation (11.46)], show that anunbiased esti-

mator for^2 is given by

11.6 G iven the data in Table 11.6:

(a) Determine the least-square estimates of and in the linear regression

equation

(b) Determine an unbiased estimate of^2 , the variance of Y.

(c) Estimate E Y at x 5.

(d) D etermine 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

Table 11.6 Data for Problem 11.6

x0123456789

y 3.2 3.1 3.9 4.7 4.3 4.4 4.8 5.3 5.9 6.0

360 Fundamentals of Probability and Statistics for Engineers

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