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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

Data:

head(cars)%>%
  kable() %>%
  kable_styling() %>% scroll_box(width = "800px")
speed dist
4 2
4 10
7 4
7 22
8 16
9 10

Exploratory Data Analysis:

plot(cars$dist, cars$speed)

Linear regression:

lmodel <- lm(dist~speed, cars)
summary(lmodel)
## 
## 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

As per the linear regression summary, speed is a good indicator of the breaking distance.

  1. The median values of the residuals are close and 1st and 3rd quadrant values are of similar magnitude.
  2. The p value is very less and indicates there is 99% probability of speed being a good indicator of breaking distance


Stopping distance = -17.5791 + 3.9324 * speed



Distributon of residuals: 

hist(lmodel$residuals)

As per the graph below there is a strong co-relation b/w speed and breaking distance, excluding some spots at the higher end of the speed where the breaking distance is disproportionately higher.

res <- resid(lmodel)
qqnorm(res)
qqline(res)