How does vehicle weight relate to fuel economy?

• Fuel economy is miles per US gallon (mpg). Higher values mean less fuel per mile.
• Weight is measured in thousands of pounds.
• We use simple linear regression to describe the association and its uncertainty.

A comparison of 2,000 lb and 3,000 lb cars corresponds to a one-unit change in weight.

• R’s built-in mtcars data contain 32 car models from 1973–74.
• Each row represents one model. We analyze wt and mpg.
• Weight ranges from 1.51 to 5.42 thousand lb.
• Fuel economy ranges from 10.4 to 33.9 mpg.

These observational data illustrate a method. They do not represent a random sample of today’s cars.

Source: R datasets::mtcars documentation, based on the 1974 Motor Trend US magazine.

The fitted line slopes downward. The shaded band shows a 95% confidence interval for the mean response under the model.

\[ Y_i = \beta_0 + \beta_1 x_i + \varepsilon_i \]

• \(Y_i\): fuel economy of car \(i\), in mpg.
• \(x_i\): weight of car \(i\), in thousands of pounds.
• \(\beta_1\): expected mpg difference per additional 1,000 lb.

Least squares chooses the line that minimizes squared residuals:

\[ (\hat\beta_0,\hat\beta_1)=\arg\min_{b_0,b_1} \sum_{i=1}^{32}\left(y_i-b_0-b_1x_i\right)^2. \]

\[ \hat\beta_1= \frac{\sum_i(x_i-\bar x)(y_i-\bar y)}{\sum_i(x_i-\bar x)^2}, \qquad \hat\beta_0=\bar y-\hat\beta_1\bar x \]

Our fitted equation is:

\[ \widehat{\mathrm{mpg}} = 37.29 - 5.34\,\mathrm{wt}. \]

An additional 1,000 lb is associated with about 5.34 fewer mpg in this dataset.

The intercept corresponds to a zero-weight car, outside the data, so it has no useful physical interpretation here.

library(ggplot2)
library(plotly)
d <- datasets::mtcars
fit <- lm(mpg ~ wt, data = d)
coef(fit)
confint(fit)
summary(fit)$r.squared

ggplot(d, aes(wt, mpg)) +
  geom_point(color = "#8C1D40") +
  geom_smooth(method = "lm", formula = y ~ x)

lm() estimates the line. confint() quantifies coefficient uncertainty. The chart lets us compare the line with the observations.

The curved residual trend suggests the straight line misses some pattern. A residual plot alone cannot verify all model assumptions.

Hover to identify models. Drag to zoom and double-click to reset. Compare cars with similar weight but different mpg.

• \(R^2 = 0.753\): weight accounts for about 75% of the observed variation in mpg around its sample mean.
• At 3,000 lb, the fitted mean is 21.3 mpg.
• Individual cars vary around that mean. A mean estimate does not guarantee the mpg of a particular car.

This describes the fitted sample. It does not establish predictive accuracy on new data.

Heavier cars in this sample tend to have lower fuel economy.

The estimated difference is about 5.3 fewer mpg per additional 1,000 lb.

Weight alone leaves meaningful differences unexplained. Engine characteristics and other design choices may matter, and the residual pattern suggests a nonlinear model could be worth checking.

These observational data support an association. They do not isolate a causal effect of adding weight, and they should not directly guide comparisons of modern cars.