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.)

library(ggplot2)

data(cars)
str(cars)
## 'data.frame':    50 obs. of  2 variables:
##  $ speed: num  4 4 7 7 8 9 10 10 10 11 ...
##  $ dist : num  2 10 4 22 16 10 18 26 34 17 ...
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

First, I started with some data exploration. I loaded the dataset and noticed which variables were observed. This data set contains 50 observations for 2 variables.

# Visualization 
ggplot(cars, aes(x = speed, y = dist)) +
  geom_point() +
  labs(title = "Scatterplot of Stopping Distance vs. Speed",
       x = "Speed",
       y = "Stopping Distance")

I started off with a visualization of the dataset. This is a scatterplot of stopping distance vs. speed. We can see that there is a trend with the data. As the speed increases, so does the stopping distance, indicating a positive relationship between the two.

model <- lm(dist ~ speed, data = cars)
print(model)
## 
## Call:
## lm(formula = dist ~ speed, data = cars)
## 
## Coefficients:
## (Intercept)        speed  
##     -17.579        3.932
ggplot(cars, aes(x = speed, y = dist)) +
  geom_point() +
  geom_line(aes(x = speed, y = fitted(model)), color = "red") +
  labs(title = "Linear Model: Speed vs. Stopping Distance",
       x = "Speed",
       y = "Stopping Distance")

This is a linear model of the regression line for this dataset. For the most part, the data follows this trend.

summary(model)
## 
## Call:
## lm(formula = dist ~ speed, data = cars)
## 
## 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

Here, we see a significant association between the two variables. The p-value of both variables in this case is less than .05. This indicates that the model is reliable for demonstrating distance as a result of speed. The R^2 value is about .65, meaning that 65% of the data’s variance is accounted for. Additionally, because the residuals are smaller values, these factors also are indicative of a normal distribution.