Using the “cars” dataset in R, build a linear model for stopping distance as a function of speed and replicate the analysis of your textbook chapter 3 (visualization, quality evaluation of the model, and residual analysis.)
summary(cars)
## speed dist
## Min. : 4.0 Min. : 2.00
## 1st Qu.:12.0 1st Qu.: 26.00
## Median :15.0 Median : 36.00
## Mean :15.4 Mean : 42.98
## 3rd Qu.:19.0 3rd Qu.: 56.00
## Max. :25.0 Max. :120.00
colnames(cars)
## [1] "speed" "dist"
plot(cars[,"speed"], cars[,"dist"], xlab="Speed", ylab="Stopping Distance", main = "Stopping distance vs Speed")
attach(cars)
cars.lm = lm(dist ~ speed)
cars.lm
##
## Call:
## lm(formula = dist ~ speed)
##
## Coefficients:
## (Intercept) speed
## -17.579 3.932
plot(speed, dist, xlab="Speed", ylab="Stopping Distance", main = "Stopping distance vs Speed")
abline(cars.lm)
summary(cars.lm)
##
## Call:
## lm(formula = dist ~ speed)
##
## Residuals:
## Min 1Q Median 3Q Max
## -29.069 -9.525 -2.272 9.215 43.201
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -17.5791 6.7584 -2.601 0.0123 *
## speed 3.9324 0.4155 9.464 1.49e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.38 on 48 degrees of freedom
## Multiple R-squared: 0.6511, Adjusted R-squared: 0.6438
## F-statistic: 89.57 on 1 and 48 DF, p-value: 1.49e-12
plot(fitted(cars.lm),resid(cars.lm), xlab="Speed", ylab="Stopping Distance Residuals", main = "Stopping distance resid vs Speed")
qqnorm(resid(cars.lm))
qqline(resid(cars.lm))
# A linear model seems to overestimate at medium speeds and underestimate at high speeds. We know from physics that Kinetic Energy (KE) = .5 * m * v^2.Work (W) = u*m*g*d. W = KE => d = v^2/(2 * u * g). So the relationship should be quadratic which we may be seeing in the qqplots.