Data was collected on 54 observations on a response of inter

Data was collected on 54 observations on a response of interest, y and four potential predictor variables x1, x2, x3 and x4. The output from regression analyses of the data is shown below.

For the best subsets regression analysis, which is the best simple linear regression model for predicting y? Briefly explain your criteria for choosing this model

For all of the models listed in the best subsets regression analysis, which model is best according to the MSE criterion?

For all of the models listed in the best subsets regression analysis, which model is best according to the BIC criterion?

Is the variable from your best simple linear regression model (from part a) included in the model with the lowest overall MSE (part b)? Briefly explain why it could happen that the best single variable is not in the best overall model.

Following the best subsets regression results, the sums of squares for regression and error (also called residual) are displayed for several models. Using the regression sums of squares information for the full model containing all four x variables, calculate i) the R² value for the full model, ii) The F statistic for the test of the H₀ : ₁ = ₂ = ₃ = ₄ = 0 and iii) the standard deviation of the residuals for the full model

Using the regression sums of squares information, test the null hypothesis H₀ : ₂ = ₄ = 0 for the full model. Calculate an F statistic, obtain a tabled F value and report the conclusion of your test, use = .05

Using the regression sum of squares information, for the model containing the terms x1, x2 and x4, calculate the t statistic for the hypothesis H₀ = ₁ = 0, where ₁ is the coefficient of x1. (Hint: first calculate an F statistic for H₀ and then take its square root to obtain the t value. Assume all regression coefficients are positive)

                                   Results for the best subsets regression analysis

                      The REG Procedure                                              Dependent variable: y

                                                    R – Square Selection method

Root

Number in Model    R – Square        C (p)           MSE               BIC           Variables in Model

         1                         0.5259          757.7545      0.19035       -180.6717             x4

         1                         0.4414         901.7010      0.20662      -171.8944             x3

         1                         0.3447          1066.354      0.22378      -163.3427             x2

1                         0.1257          1439.527      0.25849      -147.8645             x1

        2                         0.8049          284.3219      0.12329       -227.5857         x2    x3

         2                         0.6847         489.1148      0.15673      -202.2469         x3    x4

         2                         0.6521          544.6919      0.16464      -197.0205         x1    x3   

2                         0.6440          558.4141      0.16654      -195.8044         x2    x4

         3                         0.9712            3.0002        0.04781       -321.7335        x1   x2   x3

         3                         0.8758         165.5090      0.09935      -250.6043       x2   x3   x4

         3                         0.7215          428.5088      0.14878      -208.5846         x1   x3   x4

3                         0.6449          558.9234      0.16799      -195.7567         x1   x2   x4

         4                         0.9712              5.0000       0.04830      -319.5296        x1   x2   x3   x4

Results from several models to predict y using various combinations of x variables.

Terms in model                       SSRegression                   SSE or SSResidual

          x2                                       1.36979                                  2.60397

          x3                                       1.75385                                   2.21991

     x1, x3                                    2.59127                                   1.38250

     x2, x4                                    2.55926                                  1.41450

   x1, x2, x3                               3.85947                                   0.11430

x1, x2, x4                                2.56274                                   1.41103

x1, x2, x3, x4                            3.85947                                   0.11430

Solution

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For the best subsets regression analysis, which is the best simple linear regression model for predicting y? Briefly explain your criteria for choosing this model

The best model is the model that have the bigger R-square that means that when the R-square is bigger the model explain very good the \"y\" predictor

in this case there are 2 models that have the same R-square

3                         0.9712            3.0002        0.04781       -321.7335        x1   x2   x3

4                         0.9712              5.0000       0.04830      -319.5296        x1   x2   x3   x4

For all of the models listed in the best subsets regression analysis, which model is best according to the MSE criterion?

according to the MSE criterion, the best regression model is the one with the smallest MSE

In this case the best regression model will be:

3                         0.9712            3.0002        0.04781       -321.7335        x1   x2   x3