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.)

Data

data(cars)
head(cars)
##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10

Build a linear model

plot(cars$dist,cars$speed, xlab="Speed", ylab="Stop Distance")

car.lm <- lm(cars$dist ~ cars$speed)
car.lm
## 
## Call:
## lm(formula = cars$dist ~ cars$speed)
## 
## Coefficients:
## (Intercept)   cars$speed  
##     -17.579        3.932

The final regression model is: \(stopping distance = -17.579 + 3.932 \times speed\)

Evaluating the Quality of the Model

summary(car.lm)
## 
## Call:
## lm(formula = cars$dist ~ cars$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 *  
## cars$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(car.lm),resid(car.lm))
abline(0,0)

qqnorm(resid(car.lm))
qqline(resid(car.lm))

In the sacatter plot for the residules, we see that the variances of residuals areUniformly scattered about zero.

The Q-Q plot shows that the residuals roughly follow the indicated line and the residuals from the model are normally distributed.

Taken together, using speed as a predictor in the model perfectly explain the data.