library(ggplot2)
data(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
ggplot(cars, aes(x=speed, y=dist)) + 
  geom_point (color="blue") + 
  theme_light() +
  theme(plot.title = element_text(hjust = 0.5), axis.text.x = element_text(angle = 90))+
  labs(x = 'Speed, mph', y = "Stopping distance, ft", title = "Stopping Distance vs. Speed")

lm_cars <- lm(cars$dist ~ cars$speed) 
lm_cars
## 
## Call:
## lm(formula = cars$dist ~ cars$speed)
## 
## Coefficients:
## (Intercept)   cars$speed  
##     -17.579        3.932
plot(dist ~ speed, data=cars)
abline(lm(dist ~ speed, data=cars))

plot(lm_cars$residuals, pch = 16, col = "red")

summary(lm_cars)
## 
## 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

Linear Model Summary:

The residuals distribution suggests that the distribution is normal.

The standard error for the speed coefficient is ~ 9.4 (3.93/.42) times the coefficient value, which is good. The book states “For a good model, we typically would like to see a standard error that is at least five to ten times smaller than the corresponding coefficient”.

The probability that the speed coefficient is not relevant in the model is 1.49e-12 (p-value), which means that speed is very relevant in modeling stopping distance.

The p-value of the intercept is 0.0123, which means the intercept is pretty relevant in the model.

The multiple R-squared is 0.6511, which means that this model explains 65.11% of the data’s variation.

qqnorm(resid(lm_cars))
qqline(resid(lm_cars))

There is a positive correlation between the explanatory (speed) and response variable (stopping distance) and the relationship is linear. The residuals distribution also suggests that the distribution is normal.