Homework 10 Solution (Due Dec 4th, Friday)Problems:1) 9.6 Flowcharts omitted. See textbook section 9.4.2) 9.9a) See SAS output below. The maximum R2a,p,minimumCp,minimumAICp, and minimum PRESSpall identify the subset of “X1 X3” as the best subset. The plots below for R2p, Cp, AICpalso showthe same results as the table below. Note that in the plots, P = (# of predictors + intercept). Notethat the Cpvalue for “X1 X3 = 2.8, which is close to 3, the number of parameters in the model (2predictors + intercept), showing little bias. The model found by PRESSpwould be used to apply toa different data set and get “good” predictions.Number in AdjustedModel R-Square R-Square C(p) AIC Variables in Model1 0.6190 0.6103 8.3536 220.5294 X11 0.4155 0.4022 35.2456 240.2137 X31 0.3635 0.3491 42.1123 244.1312 X2---------------------------------------------------------------------------------2 0.6761 0.6610 2.8072 215.0607 X1 X32 0.6550 0.6389 5.5997 217.9676 X1 X22 0.4685 0.4437 30.2471 237.8450 X2 X3---------------------------------------------------------------------------------3 0.6822 0.6595 4.0000 216.1850 X1 X2 X3VarsInObs Model Press1 X1 5569.562 X2 9254.493 X3 8451.434 X1 X2 5235.195 X1 X3 4902.756 X2 X3 8115.917 X1 X2 X3 5057.8927b) In this case, the model that includes X1 and X3 is always preferred, but that does not need to betrue.c) No. The number of covariates is small, so all subsets does not take too long to compute.3) 9.10c) S ee SAS output below. Based on the p-values, variable X2may not be needed in the model (p-value = 0.4038). For t he correlation matrix of the predictors (see SAS output below), X2and X3aresomewhat highly correlated, and X3and X4are highly correlated.Pearson Correlation Coefficients, N = 25Prob > |r| under H0: Rho=0X1 X2 X3 X4X1 1.00000 0.10227 0.18077 0.326660.6267 0.3872 0.1110X2 0.10227 1.00000 0.51904 0.396710.6267 0.0078 0.0496X3 0.18077 0.51904 1.00000 0.782040.3872 0.0078 <.0001X4 0.32666 0.39671 0.78204 1.000000.1110 0.0496 <.0001Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel 4 8718.02248 2179.50562 129.74 <.0001Error 20 335.97752 16.79888Corrected Total 24 9054.00000Parameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr > |t|Intercept 1 -124.38182 9.94106 -12.51 <.0001X1 1 0.29573 0.04397 6.73 <.0001X2 1 0.04829 0.05662 0.85 0.4038X3 1 1.30601 0.16409 7.96 <.0001X4 1 0.51982 0.13194 3.94 0.0008284) 9.11a) See SAS output below. The four best subsets are: (X1,X3,X4), (X1,X2,X3,X4), (X1,X3) and ,(X1,X2,X3)Number in AdjustedModel R-Square R-Square C(p) AIC SBC Variables in Model1 0.8047 0.7962 84.2465 110.4685 112.90629 X31 0.7558 0.7452 110.5974 116.0546 118.49234 X41 0.2646 0.2326 375.3447 143.6180 146.05576 X11 0.2470 0.2143 384.8325 144.2094 146.64717 X2------------------------------------------------------------------------------------------2 0.9330 0.9269 17.1130 85.7272 89.38384 X1 X32 0.8773 0.8661 47.1540 100.8605 104.51716 X3 X42 0.8153 0.7985 80.5653 111.0812 114.73788 X1 X42 0.8061 0.7884 85.5196 112.2953 115.95191 X2 X32 0.7833 0.7636 97.7978 115.0720 118.72864 X2 X42 0.4642 0.4155 269.7800 137.7025 141.35916 X1 X2------------------------------------------------------------------------------------------3 0.9615 0.9560 3.7274 73.8473 78.72282 X1 X3 X43 0.9341 0.9247 18.5215 87.3143 92.18984 X1 X2 X33 0.8790 0.8617 48.2310 102.5093 107.38479 X2 X3 X43 0.8454 0.8233 66.3465 108.6361 113.51157 X1 X2 X4------------------------------------------------------------------------------------------4 0.9629 0.9555 5.0000 74.9542 81.04859 X1 X2 X3 X4b) A selection criteria such as Cp, AICp, SBCp, PRESSpmight be useful. See the output in a).5) 9.18a) See SAS output below. The final model includes (X3, X1, X4).Summary of Stepwise SelectionVariable Variable Number Partial ModelStep Entered Removed Vars In R-Square R-Square C(p) F Value Pr > F1 X3 1 0.8047 0.8047 84.2465 94.78 <.00012 X1 2 0.1283 0.9330 17.1130 42.12 <.00013 X4 3 0.0285 0.9615 3.7274 15.59 0.0007b) The mode l in a) agrees with the R2a,pcriterion obtained in Problem