Homework SourseMultiple Regression Problem Set I Using the Motor Trend Cars
Multiple Regression Problem Set I
Using the Motor Trend Cars Data Set, you would like to determine if there is a relationship between MPG (miles per gallon) and specific variables included in the data set.
a. Develop a correlation matrix using MPG (miles per gallon), HP (horsepower), WT (weight). Is the correlation between the variables significant or not? Conduct a test for all correlations; that is for MPG v. HP; MPG v. WT; HP v. WT. Use a level of significance of 0.05
b. Fit a multiple regression model using MPG (miles per gallon) as the dependent variable and HP (horsepower), and WT (weight) as the independent variables.
c. Is the overall regression model significant? Conduct the appropriate hypothesis tests for the significance of the overall regression model. Use a 0.05 level of significance.
d. Are the individual independent variables significant? Conduct a hypothesis tests for the significance of each independent variable at the= 0.05 level of significance.
e. What proportion of the variation in MPG is explained by the independent variables HP and WT?
f. Graph the residuals from the regression you developed in part (a). Do the regression assumptions appear to be appropriate given the residual plots?
data:
| mpg | cyl | disp | hp | drat | wt | qsec | vs | am | gear | carb | |
| Mazda RX4 | 21 | 6 | 160 | 110 | 3.9 | 2.62 | 16.46 | 0 | 1 | 4 | 4 |
| Mazda RX4 Wag | 21 | 6 | 160 | 110 | 3.9 | 2.875 | 17.02 | 0 | 1 | 4 | 4 |
| Datsun 710 | 22.8 | 4 | 108 | 93 | 3.85 | 2.32 | 18.61 | 1 | 1 | 4 | 1 |
| Hornet 4 Drive | 21.4 | 6 | 258 | 110 | 3.08 | 3.215 | 19.44 | 1 | 0 | 3 | 1 |
| Hornet Sportabout | 18.7 | 8 | 360 | 175 | 3.15 | 3.44 | 17.02 | 0 | 0 | 3 | 2 |
| Valiant | 18.1 | 6 | 225 | 105 | 2.76 | 3.46 | 20.22 | 1 | 0 | 3 | 1 |
| Duster 360 | 14.3 | 8 | 360 | 245 | 3.21 | 3.57 | 15.84 | 0 | 0 | 3 | 4 |
| Merc 240D | 24.4 | 4 | 146.7 | 62 | 3.69 | 3.19 | 20 | 1 | 0 | 4 | 2 |
| Merc 230 | 22.8 | 4 | 140.8 | 95 | 3.92 | 3.15 | 22.9 | 1 | 0 | 4 | 2 |
| Merc 280 | 19.2 | 6 | 167.6 | 123 | 3.92 | 3.44 | 18.3 | 1 | 0 | 4 | 4 |
| Merc 280C | 17.8 | 6 | 167.6 | 123 | 3.92 | 3.44 | 18.9 | 1 | 0 | 4 | 4 |
| Merc 450SE | 16.4 | 8 | 275.8 | 180 | 3.07 | 4.07 | 17.4 | 0 | 0 | 3 | 3 |
| Merc 450SL | 17.3 | 8 | 275.8 | 180 | 3.07 | 3.73 | 17.6 | 0 | 0 | 3 | 3 |
| Merc 450SLC | 15.2 | 8 | 275.8 | 180 | 3.07 | 3.78 | 18 | 0 | 0 | 3 | 3 |
| Cadillac Fleetwood | 10.4 | 8 | 472 | 205 | 2.93 | 5.25 | 17.98 | 0 | 0 | 3 | 4 |
| Lincoln Continental | 10.4 | 8 | 460 | 215 | 3 | 5.424 | 17.82 | 0 | 0 | 3 | 4 |
| Chrysler Imperial | 14.7 | 8 | 440 | 230 | 3.23 | 5.345 | 17.42 | 0 | 0 | 3 | 4 |
| Fiat 128 | 32.4 | 4 | 78.7 | 66 | 4.08 | 2.2 | 19.47 | 1 | 1 | 4 | 1 |
| Honda Civic | 30.4 | 4 | 75.7 | 52 | 4.93 | 1.615 | 18.52 | 1 | 1 | 4 | 2 |
| Toyota Corolla | 33.9 | 4 | 71.1 | 65 | 4.22 | 1.835 | 19.9 | 1 | 1 | 4 | 1 |
| Toyota Corona | 21.5 | 4 | 120.1 | 97 | 3.7 | 2.465 | 20.01 | 1 | 0 | 3 | 1 |
| Dodge Challenger | 15.5 | 8 | 318 | 150 | 2.76 | 3.52 | 16.87 | 0 | 0 | 3 | 2 |
| AMC Javelin | 15.2 | 8 | 304 | 150 | 3.15 | 3.435 | 17.3 | 0 | 0 | 3 | 2 |
| Camaro Z28 | 13.3 | 8 | 350 | 245 | 3.73 | 3.84 | 15.41 | 0 | 0 | 3 | 4 |
| Pontiac Firebird | 19.2 | 8 | 400 | 175 | 3.08 | 3.845 | 17.05 | 0 | 0 | 3 | 2 |
| Fiat X1-9 | 27.3 | 4 | 79 | 66 | 4.08 | 1.935 | 18.9 | 1 | 1 | 4 | 1 |
| Porsche 914-2 | 26 | 4 | 120.3 | 91 | 4.43 | 2.14 | 16.7 | 0 | 1 | 5 | 2 |
| Lotus Europa | 30.4 | 4 | 95.1 | 113 | 3.77 | 1.513 | 16.9 | 1 | 1 | 5 | 2 |
| Ford Pantera L | 15.8 | 8 | 351 | 264 | 4.22 | 3.17 | 14.5 | 0 | 1 | 5 | 4 |
| Ferrari Dino | 19.7 | 6 | 145 | 175 | 3.62 | 2.77 | 15.5 | 0 | 1 | 5 | 6 |
| Maserati Bora | 15 | 8 | 301 | 335 | 3.54 | 3.57 | 14.6 | 0 | 1 | 5 | 8 |
| Volvo 142E | 21.4 | 4 | 121 | 109 | 4.11 | 2.78 | 18.6 | 1 | 1 | 4 | 2 |
Solution
a) Develop a correlation matrix using MPG (miles per gallon), HP (horsepower), WT (weight). Is the correlation between the variables significant or not? Conduct a test for all correlations; that is for MPG v. HP; MPG v. WT; HP v. WT. Use a level of significance of 0.05
Sol) Correlation between MPG v. HP
from excel
1
Hence Correlation between MPG v. HP is-0.77617
Now Correlation between MPG v. WT
from excel
Hence Correlation between MPG v. WT is-0.8677
Now Correlation between HP v. WT.
from excel
Hence Correlation between HP v. WT. is 0.6587
b) Fit a multiple regression model using MPG (miles per gallon) as the dependent variable and HP (horsepower), and WT (weight) as the independent variables.
Y=a+b(HP)+c(WT)
By using Excel
From above anaysis
The fitted regression model is Y(MPG) = 37.227+(-0.03177)*hp+(-3.3878)* wt
c) Since P<alpha hence we can say that regression model is significant
d) P value for Independent variables
p value for hp is 0.001
p<0.05 hence hp is significant
similarly p value for wt is 0.0000012
p<0.05 hence wt is significant
e) proportion of the variation in MPG is explained by the independent variables HP and WT is 83% (Using R square value)
| mpg | hp | |
| mpg | 1 | |
| hp | -0.77617 | 1 |
