The data frame toycars consists of 27 observations on the distance (in meters) traveled by one of three different toy cars on a smooth surface, starting from rest at the top of a 16-inch-long ramp...

The data frame toycars consists of 27 observations on the distance (in meters) traveled by one of three different toy cars on a smooth surface, starting from rest at the top of a 16-inch-long ramp tilted at varying angles (measured in degrees). Because of differing frictional effects for the three different cars, we seek three regression lines that relate distance traveled to angle.

(a) As a first try, fit three lines that have the same slope but different intercepts.

(b) Note the value of R²
from the summary table. Examine the diagnostic plots carefully. Is there an influential outlier? How should it be treated?

(c) The physics of the problem actually suggests that the three lines should have the same intercept (very close to 0, in fact), and possibly differing slopes, where the slopes are inversely related to the coefficient of dynamic friction for each car. Fit the model, and note that the value of R²
is slightly lower than that for the previously fitted model. Examine the diagnostic plots. What has happened to the influential outlier? In fact, this is an example where it is inadvisable to take R²
too seriously; in this case, a more carefully considered model can accommodate all of the data satisfactorily. Maximizing R²
does not necessarily give the best model!