data-605-week11-Assignment

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.)

head(cars)

##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10

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

Let us build a linear model and find the best fitting line.

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

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

Let us look at the plot of two variables. Speed is the independent variable and stopping distance is the dependent variable.

plot(cars$speed, cars$dist, xlab='Speed (mph)', ylab='Stopping Distance (ft)', 
     main='Stopping Distance vs. Speed')
abline(cars_lm,col=c("red"))

There appears to be some correlation between two variables. Let us evaluate the linear model we have.

summary(cars_lm)

## 
## Call:
## lm(formula = cars$dist ~ cars$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 *  
## 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

The residuals distribution suggests that the distribution is normal.

The Book says “For a good model, we typically would like to see a standard error that is at least five to ten times smaller than the corresponding coefficient”.

The standard error for the speed coefficient is ~ 9.4 times the coefficient value.

The probability that the speed coefficient is not relevant in the model is 1.49e-12 (p-value), this shows the speed is very relevant in modeling stopping distiance.

The p-value of the intercept is 0.0123, which means the intercept is pretty relevant in the model.

The multiple R-squared is 0.6511, which means that this model explains 65.11% of the data’s variation.

Plot the Residuals

plot(cars_lm$fitted.values, cars_lm$residuals, xlab='Fitted Values', ylab='Residuals')
abline(0,0,col=c("red"))

It is possible to say that the outlier values do not show the same variance of the residuals.

qqnorm(cars_lm$residuals)
qqline(cars_lm$residuals)

There are some problems at the outlier levels, the normal Q-Q plot of the residuals appears to follow the theoretical line. Residuals are reasonably normally distributed.