Heavier cars seem like they would use more gas.
My question: Does a car’s weight affect its miles per gallon (mpg)?
I used the mtcars dataset that comes with R. It has 32 cars from a 1974 Motor Trend magazine.
Variables I used:
The heavier cars have lower mpg. It looks kind of like a straight line going down so linear regression should work.
Correlation tells how strong the linear relationship is between x and y.
\[r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}\]
r is always between -1 and 1. Close to -1 means a strong negative relationship.
For this data r = -0.87, so it is a strong negative relationship.
The model is
\[y = \beta_0 + \beta_1 x + \varepsilon\]
The least squares estimates are
\[\hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \qquad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}\]
To see if weight matters I tested \(H_0: \beta_1 = 0\) vs \(H_a: \beta_1 \neq 0\)
model <- lm(mpg ~ wt, data = mtcars) coef(model)
## (Intercept) wt ## 37.285126 -5.344472
confint(model)
## 2.5 % 97.5 % ## (Intercept) 33.450500 41.119753 ## wt -6.486308 -4.202635
cor(mtcars$wt, mtcars$mpg)
## [1] -0.8676594
The line fits the points pretty well. The gray area is the confidence band for the line.
You can rotate this. Cars with high weight and high horsepower have the lowest mpg.
Limitations: only 32 cars and they are all from 1974, so newer cars might be different. Also this only shows a relationship, not that weight causes lower mpg.