HW 11: Simple Linear Regression

Cars Dataset

library(tidyverse)

df <- datasets::cars

summary(df)

##      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

Visualize

plot(cars$speed, cars$dist, xlab="speed (mph)", ylab="stopping distance (ft)", main="Cars")

Build model

cars.lm <- lm(dist ~ speed, data=cars)

cars.lm

## 
## Call:
## lm(formula = dist ~ speed, data = cars)
## 
## Coefficients:
## (Intercept)        speed  
##     -17.579        3.932

\[ \widehat{dist} = - 17.579 + 3.932 * speed \] For every 1 mph increase in speed, the stopping distance increases by 3.9 feet. The intercept means that when the speed is 0 mph, the stopping distance is -17 feet, which does not make sense. The intercept does not make sense here.

plot(dist ~ speed, data=cars)
abline(cars.lm)

Model Quality Evaluation

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

• The median residual is not close to zero.
• The first and third quartiles are of similar magnitudes.
• The minimum and maximum residuals are not of similar magnitudes.
• The ratio of the speed coefficient and its standard error is 3.9324/0.4155=9.46, which means that the standard error for speed is 9.5 times smaller than the speed coefficient. This shows that there is large variability in the slope estimate.
• The p-value of the speed coefficient is very small, so we can say there is strong evidence of there being a linear relationship between speed and stopping distance.
• If the residuals are distributed normally, the first and third quartiles should be about 1.5 times the standard error. This is not true here:

\[ -9.525 \neq 1.5 * 15.38 \]

\[ 9.215 \neq 1.5 * 15.38 \]

• The multiple R-squared is 0.6511, which means that 65.011% of the variability in stopping distance is explained by the variation in the speed.
• The F-statistic isn’t too important here since the model only has one predictor.

Diagnostic plots

par(mfrow=c(2,2))
plot(cars.lm)

In the Q-Q plot, the right tail is “heavier” than what would be expected for residuals that are normally distributed. The distribution is right-skewed.