1.1 Import Data

There are 50 observations of speed of cars and the distances taken to stop from the dataset cars without any NA values.

data <- datasets::cars
head(cars)

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

dim(data)

## [1] 50  2

1.2 Visualize the Data

plot(data, main = "Stopping Dist vs Speed", xlab = "Speed (mph)", ylab = "Stopping Distance (ft)")

1.3 Build a Linear Model

Build a simple linear model with the dataset cars.

cars_lm <- lm(dist~speed, data=cars)
plot(data, main = "Stopping Dist vs Speed", xlab = "Speed (mph)", ylab = "Stopping Distance (ft)")
abline(reg=cars_lm, col="blue")

1.4 Evaluate the Model

Quality evaluation of the model:

Residuals: the median is close to zero and the 1Q vs 3Q and min vs max are roughly the same magnitude. This shows that the model is a fairly good model.

P-value: The p-value of speed is very close to zero, which shows a strong significance at 99% confidence, while the intercept is significant at 95% confidence. This shows that the model is predicting the two values quite well.

Multiple R-squared: The \(R^{2}\) value represents how well the model describes the measured data. It explains 65.11% of the data’s variation, which is fairly good.

summary(cars_lm)

## 
## 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

1.5 Residual Analysis

The residuals vary around and are not uniformly scattered above and below zero. They do not show any special patterns.

plot(fitted(cars_lm), resid(cars_lm), 
     main = "Cars: Fitted Stopping Distance vs Residuals", 
     xlab = "Fitted Stopping Distance", ylab = "Residuals")
abline(h=0, col="blue")

The QQ plot shows some outliers above the line at the upper end and the lower end. The residuals might not be very normal.

qqnorm(resid(cars_lm))
qqline(resid(cars_lm))

To check the normality, we can also look at the histogram or a normal probability plot of the residuals.

The histogram is bimodal and slightly right-skewed. Therefore, the residuals are barely normal.

hist(resid(cars_lm), breaks=15, prob=TRUE)
curve(dnorm(x, mean=mean(resid(cars_lm)), sd=sd(resid(cars_lm))), col="red", add=TRUE)