Question

Construct a higher order models until you are satisfied you have found the best polynomial model.

1. Write out your final model or display the analysis window for it so I can see the model you settled on.
2. What is R2-adjusted?
3. Include the Plot of Fitted Model and Residuals versus X output.
4. Predict the highway mileage of a car with 150 horsepower. Also construct 95% confidence and prediction intervals for such a car.
5. Horsepower   MPG Highway
   140   31
   200   25
   172   26
   172   26
   208   30
   110   31
   170   28
   180   25
   170   27
   200   25
   295   25
   110   36
   110   34
   160   28
   110   29
   170   23
   165   20
   170   26
   300   25
   153   28
   141   28
   147   26
   92   33
   93   29
   100   27
   142   21
   100   27
   300   24
   92   33
   214   28
   63   33
   127   30
   96   27
   105   29
   115   30
   145   20
   140   30
   190   26
   55   50
   90   36
   160   31
   102   46
   140   31
   81   33
   124   29
   92   34
   128   27
   278   22
   185   24
   225   23
   160   26
   210   26
   82   37
   103   36
   164   34
   155   24
   255   25
   130   29
   217   25
   100   26
   140   26
   92   33
   202   24
   110   33
   150   30
   151   23
   160   26
   155   31
   110   31
   170   23
   170   28
   92   30
   74   41
   110   31
   160   28
   200   27
   170   28
   140   26
   85   38
   73   37
   90   30
   130   30
   70   43
   82   37
   135   32
   130   29
   138   22
   81   33
   109   21
   134   30
   178   25
   114   28
   168   28
   

Answer 1

a)

Regression Analysis: MPG Highway versus Horsepower The regression equation is MPG Highway = 38.1 -0.0630 Horsepower Predictor Coef Constant 38.150 Horsepower -0.063019 SE Coef 1.282 29.76 0.000 0.008381 -7.52 0.000 - 4.21025 R-Sq = 38.3% R-Sq(adj) = 37.6% Regression Analysis: MPG Highway versus Horsepower, Horsepower^2 The regression equation is MPG Highway - 50.2 - 0.225 Horsepower + 0.000482 Horsepower-2 Predictor Coef SE Coef T P Constant 50.196 2 .728 18.40 0.000 Horsepower -0.225210 .03416 -6.590 .000 Horsepower2 0.00048209 0.00009906 4.87 0.000 S = 3.76685 R-Sq = 51.2 R-Sq(adj) = 50.13 Regression Analysis: MPG Highway versus Horsepower, Horsepower^2, ... The regression equation is MPG Highway - 69.0 - 0.617 Horsepower + 0.00293 Horsepower-2 - 0.000005 Horsepower 3 Predictor Constant Horsepower Horsepower 2 Horsepower-3 Coef 68.984 -0.6172 0.0029265 -0.00000458 SE CoefT P 7.171 9.62 0.000 0.1431 -4.31 0.000 0.0008733 3.35 0.001 0.00000163 -2.82 0.006 S = 3.62972 R-Sq = 55.2: R-Sq(adj) = 53.73

From the above three polynomial models with degrees 1, 2 and 3 respectively on the explanatory variable  Horsepower, the corresponding R-sq(adj) are 37.6%, 50.1%, and 53.7% respectively. The increase in R-sq(adj) from degree 1 to degree 2 model is approximately 10 with an increasing 1 parameter on the model. Hence, the model 2 with degree 2 fits better than model 1 with degree 1. The increase in R-sq(adj) from degree 2 to degree 3 model is approximately 4 with an increasing 1 parameter on the model. Hence, the model 3 with degree 3 does not increase the fitness fits than model 2. Hence, the best polynomial model for these given data is the model with degree 2.

b) The R2-adjusted= R-Sq(adj) = 50.1%

c)  the Plot of Fitted Model and Residuals versus X output is

Scatterplot of Residuals vs Fitted Residuals 25.0 27.5 30.0 32.5 Fitted 35.0 37.5 40.0

From the above residual VS fitted value plot, the plot shows that the residuals and the fitted values are uncorrelated. Also, the linear model should be in a homoscedastic with normally distributed errors. Whereas, some of the points (around three points) have deviated largely from the remaining. Hence, some data points may be the outlier.

d) Predict the highway mileage of a car with 150 horsepower. Also, construct 95% confidence and prediction intervals for such a car.

Ans: The predicted highway mileage of a car with 150 horsepower is

MPG Highway = 50.2 - 0.225* 150 + 0.000482 *150^2=  27.262.

95% confidence intervals for such a car is  (26.286, 28.239).

95% prediction intervals for such a car is  (19.715, 34.809).