Numerical Questions Answer the following using the R statistical computing platform. Your answer should include the code you wrote, the output of such code, and written, English coding comments where necessary. The attached data set found in "Topic 7 Assignment Data" shows mile per gallon {variable to be regressed] and the cars' specifications [explanatory] variables. 1. Does the correlation matrix give any indication of multicollinearity? Calculate the variance inflation factors and the condition number of X'X. Is there any evidence of multicollinearity? Find the eigenvectors associated with the smallest eigenvalues of X'X. Interpret the elements of these vectors. What can you say about the source of multicollinearity in these dataEI 2. Examine the patient satisfaction data from Topic 2 for multicollinearity. Perform a principle component regression, and find the variance inflation factors. What can you conclude? Theoretical Questions Recall the principalcomponent regression model J! = Zn! + E, where Z = KT,\" = T', T'X'XT = 212 = . 1. Show that if XIX is in correlation form, A is the diagonal matrix of eigenvalues of XX, and T is the corresponding matrix of eigenvectors, then the variance inflation factors are the main diagonal elements of Tit1'1". Fuel Efficiency (g/km) Cetane Number Density Viscosity Initial Boiling Pt Final Boiling Pt Flash Pt Total Aroma 343 52.8 811.7 2.11 228 261 87 1.8 356 52.8 811.7 2.11 228 261 87 1.8 344 58 821.3 2.11 223 268 87 16.6 356 58 821.3 2.11 223 268 87 16.6 352 47.2 832 2.89 221 261 92 23 361 47.2 832 2.89 221 261 92 23 372 47 831.3 2.26 198 323 75 25.1 355 47 831.3 2.26 198 323 75 25.1 375 48.3 836.8 2.47 188 364 71 26.1 359 48.3 836.8 2.47 188 364 71 26.1 364 44.7 888.3 1.41 188 388 64 28 357 44.7 888.3 1.41 188 388 64 28 368 55.7 888.7 1.44 176 299 64 28.5 368 55.7 888.7 1.44 176 299 64 28.5 372 52.8 813.2 1.96 175 381 75 17.3 352 52.8 813.2 1.96 175 381 75 17.3