The data was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles. (“http://stat.ethz.ch/R-manual/R-devel/library/datasets/html/mtcars.html”)

Purpose: fit a linear model to predict the impact of car weight on cars’ fuel cost

  1. Scatterplot
attach(mtcars)

plot(wt, mpg, main="Relationship between Car Weight and Miles Per Gallon", 
     xlab="Car Weight", ylab="Miles Per Gallon", pch=19)

We can observe a negative relationship between car weight and fuel coast. It is close to linear relationship.

  1. Regression Model
results=lm(mpg~wt, data=mtcars)

summary(results)
## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

I fit a linear model to predict the impact of car weight on miles per gallon. The result suggests that the coefficient p-value of car weight is statistically significant (p<0.001). The value of the coefficient suggest that car weight have a negative relationship with fuel cost. The value of R-squared (0.7528) indicates that 75.28% of variance in the fuel cost can be explained by car weight.

  1. Residual Histogram
summary(results$residuals)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -4.5432 -2.3647 -0.1252  0.0000  1.4096  6.8727
hist(results$residuals, main="Residual Histogram", xlab="Residuals")

Histogram of residuals is used to detect violation of normality assumption. The distribution of residuals is close to normal distribution. Overall, the results suggest a good model fit.