Data 605 Assignment 11

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.)

# cars data
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

visualization

carslm <- lm(cars$dist ~ cars$speed)
plot(cars$speed, cars$dist, main = "Comparsion between speed and distance", xlab = "speed", ylab="distance")
abline(carslm)

Linear Regression Model

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

It is a strong uphill (positive) linear relationship.

# linear regression model
carslm <- lm(cars$dist ~ cars$speed)
summary(carslm)
## 
## 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

When Y is stop distance and X is spreed, the linear model is as follow: \(Y = -17.5791 + 3.9324 X\)

It is a a statistically significant predictor of evaluation score with p-value less than 0.05. For Multiple R-squared, the model is around 65% fits the data.

# histogram of residuals
hist(carslm$residuals)

Histogram of residual plot appear to be near normally distributed.

# qqplot
qqnorm(carslm$residuals)
qqline(carslm$residuals)

Q-Q plot are not uniformly scattered and have deviation at lower and quantiles. The residuals does not show randomly.