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.)

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
lm_cars <- lm(cars$dist ~ cars$speed)

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

Visualization:

plot(cars$speed, cars$dist, xlab = "Speed (mph)", ylab = "Distance", col='blue')
abline(lm_cars, col = "red")

Residuals:

residual <- resid(lm_cars)

summary(residual)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -29.070  -9.525  -2.272   0.000   9.215  43.200
hist(residual, xlab = "Residuals of Distance")

Residual is right skewed.

plot(fitted(lm_cars), resid(lm_cars),  col = 'blue')

qqnorm(residual)
qqline(residual)

Using Log

lm2 <- lm(log(cars$dist) ~ log(cars$speed))
summary(lm2)
## 
## Call:
## lm(formula = log(cars$dist) ~ log(cars$speed))
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.00215 -0.24578 -0.02898  0.20717  0.88289 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      -0.7297     0.3758  -1.941   0.0581 .  
## log(cars$speed)   1.6024     0.1395  11.484 2.26e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4053 on 48 degrees of freedom
## Multiple R-squared:  0.7331, Adjusted R-squared:  0.7276 
## F-statistic: 131.9 on 1 and 48 DF,  p-value: 2.259e-15
plot(log(cars$speed), log(cars$dist), xlab = 'log: Speed ', ylab = 'log: Distance', col = 'blue')
abline(lm2, col = 'red')

residual2 <- resid(lm2)

summary(residual2)
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -1.00200 -0.24580 -0.02898  0.00000  0.20720  0.88290
hist(residual2, xlab = "Residuals of Distance")

qqnorm(residual2)
qqline(residual2)

Using Log, Residuals appear to be normal and hetroskedacity is improved.