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 Visualization
# Summary statistics of the dataset
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
# Identify a linear relationship between independent and the response variables
plot(cars$speed, cars$dist, main = "Stopping Distance vs Speed", xlab = "Speed", ylab = "Distance")

Modeling
# Creating the linear regression model
cars_lm <- lm(cars$dist ~ cars$speed)

# Regression Model: Stopping Distance = -17.579 + 3.932*Speed
cars_lm
## 
## Call:
## lm(formula = cars$dist ~ cars$speed)
## 
## Coefficients:
## (Intercept)   cars$speed  
##     -17.579        3.932
plot(cars$speed, cars$dist, main = "Stopping Distance vs Speed (with Fitted Regression Line)", xlab = "Speed", ylab = "Distance")
abline(cars_lm, col="blue")

Model Evaluation
summary(cars_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$speed    3.9324     0.4155   9.464 1.49e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.38 on 48 degrees of freedom
## Multiple R-squared:  0.6511, Adjusted R-squared:  0.6438 
## F-statistic: 89.57 on 1 and 48 DF,  p-value: 1.49e-12
# This looks like a good model as the residuals mean is close to zero, min/max and 1Q/3Q are roughly the same magnitude.
# The Std Error of the speed variable is 9 times smaller than the calculated coefficient, which is good (expectation is between 5 & 10x)
# R_squared is 0.6511 which describes the model's ability to explain over 65% of the data variability
Residuals Analysis
plot(cars_lm$fitted.values, cars_lm$residuals, xlab='Fitted Values', ylab='Residuals')
abline(0,0, col="red")

qqnorm(cars_lm$residuals)
qqline(cars_lm$residuals)

# Residuals plot shows a relatively constant variability with no apparent patterns
# Q-Q plot shows the residuals relatively following the theoretical straight line (except on the ends), which denotes a normal distribution
Conclusion
The linear regression model looks relatively good and appears to describe about 65% of the data variability. This aligns with the reasonable expectation that the stopping distnce is linearly (positively) correlated with the speed