Question

Perform another multiple regression model for gasoline mileage as it depends on the set of independent variables retained from part 12.e (use the same definitions of the variables as you used in part 1).

• Present a copy of this regression output report.

a.​State (here) this multiple regression equation.

b.​Overall, is this regression significant?

c.​Comment on the significance of each individual variable coefficient.

d.​How much of the variation in mpg is fairly represented by this model?

MPG 43.1 19.9 19.2 Horsepower 48 110 105 165 139 103 115 155 142 150 71 76 65 100 84 58 88 92 139 110 90 17.7 18.1 20.3 21.5 16.9 ISS 185 27.2 41.5 46.6 23.7 27.2 39.1 28.0 24.0 20.2 20.5 28.0 34.7 36.1 35.7 20.2 23.9 29.9 30.4 36.0 22.6 36.4 27.5 33.7 44.6 32.9 38.0 24.2 38.1 39.4 25.4 31.3 34.1 34.0 31.0 27.4 223 28.0 17.6 34.4 20.6 Weight 16RS 3365 3535 3445 3205 2830 3345 4360 4054 3940 2190 2044 2110 2420 2490 1455 2305 2465 3570 3155 2678 2015 1800 2015 2965 2920 2380 2850 1980 2800 2950 2560 1910 66 80 85 90 65 67 74 110 87 95 75 Transmission manual manual manual manual manual manual automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic manual manual manual manual manual manual manual manual automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic manual manual manual manual manual manual 1850 100 67 120 60 70 116 75 2615 1963 3230 1968 2070 2985 1842 1985 2395 2720 2470 2890 2025 3465 3465 3380 82 80 79 85 105

a.​Present the plot of the residuals of this multiple regression model against its fitted values.

​b. Describe the appearance of this residual plot.

c.​State (here) the RMSE of this regression.

Answer 1

Perform another multiple regression model for gasoline mileage as it depends on the set of independent variables retained from part 12.e

To do the regression equation we do the following steps in MINITAB:

1. Enter the given values in different columns.

2. Go to “Stat” then “Regression” then “Regression” then “Fit regression model”.

3. Enter MPG in “response” and HorsePower, Weight, Transmission in “continuous predictor”.

4. Click on Graphs and check the residual versus fit graph

5. Click OK.

Thus we get the following output:

MPG = 55.19 - 0.1404 Horsepower - 0.00471 Weight - 3.35 Transmission

Analysis of Variance

Source          DF Adj SS Adj MS F-Value P-Value
Regression       3 2409.1 803.04    42.82    0.000
Horsepower     1   224.9 224.86    11.99    0.001
Weight         1   154.3 154.30     8.23    0.006
Transmission   1   121.9 121.90     6.50    0.014
Error           46   862.7   18.75
Total           49 3271.8

Model Summary

      S    R-sq R-sq(adj) R-sq(pred)
4.33070 73.63%     71.91%      67.46%

Coefficients

Term              Coef SE Coef T-Value P-Value   VIF
Constant         55.19     2.51    22.01    0.000
Horsepower     -0.1404   0.0405    -3.46    0.001 3.17
Weight        -0.00471 0.00164    -2.87    0.006 3.25
Transmission     -3.35     1.31    -2.55    0.014 1.08

Regression Equation

MPG = 55.19 - 0.1404 Horsepower - 0.00471 Weight - 3.35 Transmission

Fits and Diagnostics for Unusual Observations

                            Std
Obs    MPG    Fit Resid Resid
4 17.70 12.47   5.23   1.45     X
13 46.60 36.14 10.46   2.48 R
49 34.40 26.41   7.99   2.18 R X

R Large residual
X Unusual X

a) The regression equation is:

MPG = 55.19 - 0.1404 Horsepower - 0.00471 Weight - 3.35 Transmission

b) At 0.05 significance level we consider the the regression equation to be significant if p value <=0.05. Here p value = 0.000 is less than 0.05 so we conclude that the overall regression is significant.

c) Here we have to comment on the significance of each individual variable coefficient. At 0.05 significance level we consider the coefficient to be significant if p value <=0.05.

Here p value of Horsepower = 0.001 , p value of weigth = 0.006 an dp value of transmission = 0.014 which are all less than 0.05 so we conclude that all the 3 variables are significant.

d) Here we see 73.63% which is the value for r so 73.63% of the variation in MPG is fairly represented by this model.

a) ​We have to present the plot of the residuals of this multiple regression model against its fitted values. Running the above steps we get:

Versus Fits (response is MPG) 10 5 Residual -5 -10 10 15 20 30 35 40 25 Fitted Value

b) From the given plot we see there is a random pattern which indicates that a linear relationship is present with a moderately good fit.

c) From the output we see that the MSE= 18.75.

So $RMSE= \sqrt{MSE}=\sqrt{18.75}=4.3301$