Question

6. (textbook) An analyst fitted a regression model to predict city MPG using as predictors Length (of car in inches), Width (of car in inches) and Weight (of car in pounds). a. Intuitively, what association do you expect between the explanatory variables and MPG? b. Do you see anything of concern about these variables being used as explanatory variables? Explain S c. What does the matrix plot done in class show you? Explain d. Write the null and alternative hypothesis that checks the regression model on all explanatory variables According to the output shown below, was the regression fit statistically significant? (was the null rejected?) e. Analysis of Variance DF SS MS F P Source Regression 3 547.37 182.46 27.90 0.000 Residual Error 39 255.09 6.54 Total 42 802.47 I Predictor Constant Length Width Weight Coef SE Coef 39.449 -0.00157 0.0454 -0.03 0.973 -0.0463 0.1373 -0.34 0.738 -0.0043 0.000839 -5.17 0.000 T P 8.168 4.83 0.000 65.8% S=2.55751 R-Sq 68.2% R-Sq(adj) f. Which would be your final model (regression equation) to predict city MPG? Explain g. Using your FINAL model, what would be the MPG if the length of the weight is 3600 pounds? car is 80 inches, the width is 60 inches and the h. What would be the effect on MPG of increasing the weight by 1 pound? What about increasing the length by 1 inch? i. According to the diagnostics do you see anything to be concerned about?

Answer 1

Solution

Back-up Theory

p-value < significance level => significance ................................................................................... (1)

Beta coefficients in the regression equation represents the change in the response variable

per change of 1 uint in the respective predictor variable...................................................................(2)

Now, to work out the solution,

a) MPG is likely to have a negative impact from weight.     Length and width can have hardly any impact on MPG. Answer 1

b) Yes; Weight is dependent on the length and width and hence the model has to take account of multicollinearity. Answer 2

c) Cannot address the question since matrix plot referred in the question does not appear in the question. Answer 3

d) H01: β1 = 0 Vs H11: β1 ≠ 0; H02: β2 = 0 Vs H12: β2 ≠ 0; H03: β3 = 0 Vs H13: β3 ≠ 0. Answer 4

e) Yes: p-value is insignificantly low. Answer 5 [vide (1)]

f) Final model would be: MPG = α + βWeight; Because, from the output immediately following ANOVA Table,     the p-value is significantly large for length and width indicating that these two variables are not significant;    while p-value for constant and weight are insignificantly low implying a high significance. Answer 6    [vide (1)]

g) Regression equation would be: MPG = 39.449 - 0.0043 Weight. Substituting W = 3600, MPG = 23.969 Answer 7

h) Increasing weight by 1 pound would bring down the MPG by 0.0043.     Since length is insignificant, MPG would not be affected. Answer 8 [vide (2)]

i) Yes; As already mentioned under (b), problem of multi-collinearity needs to be accounted for. Answer 9

DONE