Question

The Motor Trend Car Road Tests dataset mtcars, in faraway R package, was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973–74 models). The data frame has 32 observation on 11 (numeric) variables: mpg: Miles/(US) gallon; cyl: Number of cylinders; disp: Displacement (cu.in.); hp: Gross horsepower; drat: Rear axle ratio; wt: Weight (1000 lbs); qsec: 1/4 mile time; vs: Engine (0 = V-shaped, 1 = straight); am: Transmission (0 = automatic, 1 = manual); gear: Number of forward gears, and carb: Number of carburetors. Use mpg as the response variable. Fit a regression model with the mpg: Miles/(US) gallon as the response and the other variables as predictors. (a) What percentage of variation in the response is explained by these predictors? (3) STA3701/101/3/2019 (b) Which observation has the largest (positive) residual? Give the case number. (2) (c) Compute the mean and median of the residuals. (2) (d) Compute the correlation of the residuals with the fitted values. Interpret the result. (3) (e) Compute the correlation of the residuals with the cyl: Number of cylinders. What does the result imply? (3) (f) For all other predictors held constant what would be the difference in predicted Miles/(US) gallon for an automobile with automatic transmission compared to a manual transmission?

Answer 1

Fit the regression model in R by the below command.

model = lm(mpg ~ . , data = mtcars)

Store the summary of the model.

s = summary(model)

(a) What percentage of variation in the response is explained by these predictors?

Run the command s$r.squared

> s$r.squared
[1] 0.8690158

86.90% of variation in the response is explained by these predictors

b) Which observation has the largest (positive) residual? Give the case number.

Run the command which(s$residuals == max(s$residuals)) to get the observation with the largest (positive) residual.

> which(s$residuals == max(s$residuals))
Fiat 128
18

Fiat 128 (18th observation) has the highest positive residuals.

(c) Compute the mean and median of the residuals.

Mean and median of the residuals are 7.459311e-17 and -0.1196051

> mean(s$residuals)
[1] 7.459311e-17
> median(s$residuals)
[1] -0.1196051

(d) Compute the correlation of the residuals with the fitted values. Interpret the result.

Run the command cor(s$residuals, model$fitted.values) to get the correlation.

> cor(s$residuals, model$fitted.values)
[1] -1.557446e-16

Since the correlation of residuals and fitted values is almost 0, residuals and fitted values are independent of each other. This implies that the error term is independent of the fitted values.

(e) Compute the correlation of the residuals with the cyl: Number of cylinders. What does the result imply?

> cor(model$residuals, mtcars$cyl)
[1] -1.857654e-16

Since the correlation of residuals and cyl is almost 0, residuals and cyl are independent of each other.

(f) For all other predictors held constant what would be the difference in predicted Miles/(US) gallon for an automobile with automatic transmission compared to a manual transmission?

In the below output of the estimated linear regression, the estimated coefficient of am is 2.52023.

> model

Call:
lm(formula = mpg ~ ., data = mtcars)

Coefficients:
(Intercept) cyl disp hp drat wt qsec vs
12.30337 -0.11144 0.01334 -0.02148 0.78711 -3.71530 0.82104 0.31776
am gear carb
2.52023 0.65541 -0.19942

Thus, the difference in predicted Miles/(US) gallon for an automobile with automatic transmission compared to a manual transmission is 2.52023.