HW11

Stopping Distance and Speed

Summary of Cars Data

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
cor(cars$dist,cars$speed)
## [1] 0.8068949

Looking at the data of the cars dataset and running the cor function we see that there is a high correlation between the speed and stopping distance.

Linear Model

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

summary(cars_lm)
## 
## 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

Looking at the summary of the linear model we see that there is a R-squared of .6511 meaning that this model explains around 65% of the data’s variation.

Our standard error for speed (0.4155) is 9.5 times less than the coefficient value (3.9324) which is good since a good model has the standard error that is at least 5 to 10 times smaller than the coefficient.

plot(cars$speed, cars$dist, xlab = "Speed (mph)", ylab = "Stopping Distance (ft)",main="Relationship Between Speed and Stopping Distance")
abline(cars_lm)

Our least squares regression line is:
\[stopping distance = -17.58 + 3.93(speed)\]

Residuals

plot(cars_lm$residuals ~ cars$speed, xlab='Fitted Values', ylab='Residuals',main="Speed vs Linear Model Residuals")
abline(h=0, lty=3)

Our residuals look to be uniformally scattered above and below 0. This says that speed is factor in the stopping distance and can explain the data.

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

The Q-Q plot also shows that speed is a factor in stopping distance as it is almost in a straight line.