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).

Libraries

library(tidyverse)
## Warning: package 'tidyverse' was built under R version 3.4.3
## -- Attaching packages ---------------------------------- tidyverse 1.2.1 --
## v ggplot2 2.2.1     v purrr   0.2.4
## v tibble  1.4.1     v dplyr   0.7.4
## v tidyr   0.7.2     v stringr 1.2.0
## v readr   1.1.1     v forcats 0.2.0
## Warning: package 'tibble' was built under R version 3.4.3
## Warning: package 'tidyr' was built under R version 3.4.3
## Warning: package 'readr' was built under R version 3.4.3
## Warning: package 'purrr' was built under R version 3.4.3
## Warning: package 'dplyr' was built under R version 3.4.2
## Warning: package 'forcats' was built under R version 3.4.3
## -- Conflicts ------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

Take a look at the cars dataset with the glimpse function from the tidyverse

glimpse(cars)
## Observations: 50
## Variables: 2
## $ speed <dbl> 4, 4, 7, 7, 8, 9, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13...
## $ dist  <dbl> 2, 10, 4, 22, 16, 10, 18, 26, 34, 17, 28, 14, 20, 24, 28...

We see that we have 2 variables, (speed, dist) and 50 observations.

First, using the summary feature in R we will view several descriptive statistics.

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

According to https://www.youtube.com/watch?v=KsVBBJRb9TE&feature=youtu.be the x variable is the explanatory variable, while the y variable is the response variable. In this case, the explanatory variable is speed while the response variable is dist

Next we will plot the two variables

plot(cars$speed, cars$dist,
     xlab="Speed (MPH)",
     ylab="Distance FT (Stopping)",
     main="Stopping Distance vs. Speed",
     col="blue")

Using the lm() function we will create the relationship model between the variables.

cat("The Relationship Model Between Speed & Distance","\n")
## The Relationship Model Between Speed & Distance
(lm_cars <-lm(cars$dist ~ cars$speed))
## 
## Call:
## lm(formula = cars$dist ~ cars$speed)
## 
## Coefficients:
## (Intercept)   cars$speed  
##     -17.579        3.932

Visualize the Regression In this plot we add the abline which is based off of our findings from lm_cars

plot(cars$speed, cars$dist,
     xlab="Speed (MPH)",
     ylab="Distance FT (Stopping)",
     main="Stopping Distance vs. Speed",
     col="blue")
abline(lm_cars)

Next, using the summary feature in R we will view the descriptive statistics for lm_cars

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

Residual Analysis

plot(lm_cars$fitted.values, lm_cars$residuals,
     xlab = "Fitted Values",
     ylab = "Residuals")
abline(0,0)