Fundamentals of Probability and Statistics for Engineers

360 Fundamentals of Probability and Statistics for Engineerswhere w i are assigned weights. In vector–matrix notation, show that estimates^ and ^ now take the form^q ˆ ^^ˆ…C T WC† 1 C T Wy;where23w 1 0w 2 W ˆ 6 . .74. 5 :0 w n11.5 (a) In simple linear regression [Equation (11.4)]. use vector–matrix notation andshow that the unbiased estimator for 2 given by Equation (11.33) can bewritten in the form(b) In multiple linear regression [Equation (11.46)], show that an unbiased estimatorfor 2 is given by11.6 Given the data in Table 11.6:c 2 ˆ 1n 2 ‰…Y C ^Q† T …Y C ^Q†Š:c 2 ˆ1n m 1 ‰…Y C ^Q† T …Y C ^Q†Š:Table 11.6 Data for Problem 11.6x 0 1 2 3 4 5 6 7 8 9y 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 regressionequationY ˆ ‡ x ‡ E:(b) Determine an unbiased estimate of 2 , the variance of Y .(c) Estimate E fY g at x ˆ 5.(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 measurementstaken for a class of light vehicles are given in Table 11.7. Assuming thatY ˆ ‡ log 10 v ‡ E;TLFeBOOK