1. 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).

EDA

Let’s start our analysis with a little exploratory data analysis to ensure we understand the data set. We’ll the use the str and summary functions for starts.

## 'data.frame':    50 obs. of  2 variables:
##  $ speed: num  4 4 7 7 8 9 10 10 10 11 ...
##  $ dist : num  2 10 4 22 16 10 18 26 34 17 ...
##      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

Now let’s plot our variables: speed(explanatory) and and stopping distance (response).

Model Building and Evaluation

Next we will build our linear model lm(cars$dist ~ cars$speed)

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

There appears to be a linear relationship between two variables - let’s see how the model performed.

## 
## 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

Both the intercept and coefficient are significant, thought the coefficient is more so. The R-squared is a very respectable 65.11% and the adjusted R-Squared is 64.38%. The F-statistics was 89.57.

The plot of the residuals appear scattered fairly evenly around zero, with a median close to zero of -2.7.

The QQ plots indicate some skew, hovever, overall they reflect a near-normal distribution for the residuals.

Conclusion

The model’s coefficient, R-squared and F statistic indicate the model does a fairly good job capturing the relationship between speed and stopping. Adding additional variables, however, could improve performance - as the R-squared is explaining approximately 65% of the variance. However, for a simple one variable linear model the model does well.