Using the gasoline mileage data in Table B. 3 find the eigenvectors associated with the smallest eigenvalues of \(\mathbf{X}^{\prime} \mathbf{X}\). Interpret the elements of these vectors. What can you say about the source of multicollinearity in these data?

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Automobile y X2 X3 X4 Xs X6 X7 8 Xg X10 X11 Apollo 18.90 350 165 260 8.0:1 2.56:1 Omega 1700 350 170 275 8.5:1 2.56:1 4 +4 Nova Monarch 20.00 250 18.25 351 105 185 8.25:1 2.73:1 1 143 255 Duster 20.07 225 Jenson 11.2 440 8.0:1 95 170 8.4:1 215 330 8.2:1 3.00:1 2 2.76:1 1 3 2.88:1 4 3 3 200.3 69.9 3910 A 3 199.6 72.9 2860 A 3 196.7 72.2 3510 A 3 199.9 74.0 3890 A 194.1 71.8 3365 M 184.5 69 4215 A Conv. Skyhawk 22.12 231 110 175 Monza Scirocco 21.47 262 34.70 89.7 Corolla 30.40 96.9 8.0:1 110 200 8.5:1 70 81 8.2:1 75 83 2.56:1 23 2.56:1 23 179.3 65.4 3020 A 179.3 65.4 3180 A 3.90:1 2 4 155.7 64 1905 M 9.0:1 4.30:1 25 165.2 65 2320 M SR-5 Camaro 16.50 350 Datsun 36.50 85.3 155 250 80 83 8.5:1 3.08:1 8.5:1 3.89:1 2 42 34 195.4 74.4 3885 A 160.6 62.2 2009 M B210 Capri II 21.50 171 109 146 8.2:1 3.22:1 Pacer 19.70 258 110 195 8.0:1 3.08:1 Babcat 20.30 140 Granada 17.80 302 83 129 220 8.0:1 109 8.4:1 3.40:1 3.0:1 Eldorado 14.39 500 190 360 8.5:1 2.73:1 Imperial Nova LN 14.89 440 215 330 8.2:1 2.71:1 Valiant 17.80 16.41 318 350 155 250 8.5:1 3.08:1 Starfire 23.54 231 145 255 8.5:1 110 175 2.45:1 8.0:1 2.56:1 Cordoba 21.47 360 180 290 8.4:1 2.45:1 2 Trans AM 16.59 400 185 NA 76:1 3.08:1 Corolla E-5 31.90 96.9 75 83 9.0:1 4.30:1 Astre 29.40 140 86 NA 8.0:1 2.92:1 Mark IV 13.27 460 Celica GT 23.90 133.6 Charger SE 19.73 318 223 366 8.0:1 96 120 8.4:1 140 255 8.5:1 3.00:1 4 3.91:1 2.71:1 2122444222422422 2 2 25 2 3 5453333333333343 170.4 66.9 2655 M 171.5 77 3375 A 168.8 69.4 2700 M 199.9 74 3890 A 224.1 79.8 5290 A 231.0 79.7 5185 A 179.3 65.4 3 214.2 76.3 196 73 196.7 72.2 3910 A 1976 71 3660 A 3050 A 4250 A 3850 A Cougar 13.90 351 Elite Matador Corvette 13.27 351 13.77 360 16.50 350 148 243 8.0:1 3.25:1 2 148 243 8.0:1 3.26:1 2 195 295 8.25:1 3.15:11 165 255 8.5:1 2.73:1 2244 3 165.2 61.8 2275 M 176.4 65.4 2150 M 228 79.8 5430 A 171.5 63.4 2535 M 215.3 76.3 4370 A 215.5 78.5 4540 A 333 216.1 78.5 4715 A 209.3 77.4 4215 A 185.2 69 3660 A