How does fuel economy vary with car weight?

• Data: 32 cars, 1973-74 models, in R’s built-in mtcars dataset.
• Predictor: wt, weight in thousands of pounds.
• Response: mpg, miles per US gallon.
• These observational data illustrate a method. They do not represent today’s entire car market.

Source: R datasets documentation: mtcars.

The line summarizes the trend. Shading shows a 95% confidence interval for the mean response under the model.

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

• \(\beta_0\): mean response at \(x=0\).
• \(\beta_1\): change in the mean response for a one-unit increase in \(x\).
• \(\varepsilon_i\): variation around the line, with \(E(\varepsilon_i\mid x_i)=0\).

Here, one unit of \(x\) means 1,000 pounds. The fitted line estimates mean mpg at a given weight.

\[\hat{\beta}_1 = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})} {\sum_{i=1}^{n}(x_i-\bar{x})^2}, \qquad \hat{\beta}_0 = \bar{y}-\hat{\beta}_1\bar{x}\]

\[\operatorname{SSE} = \sum_{i=1}^{n}(y_i-\hat{y}_i)^2\]

Least squares chooses the intercept and slope that minimize SSE.

A residual is the vertical difference between an observed mpg value and the fitted line.

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

• An additional 1,000 lb is associated with 5.34 fewer mpg, on average, in this fitted model.
• \(R^2 = 0.753\): weight accounts for about 75.3% of the observed variation in mpg around its sample mean.
• A zero-weight car lies outside the data. The intercept has no useful physical interpretation here.

\[\hat{y} = 37.2851 - 5.3445(3) \approx 21.25\ \mathrm{mpg}\]

The prediction interval is wider because individual cars vary around the mean. Both intervals depend on the regression assumptions.

The curved smooth suggests the straight line misses some structure. Several cars differ from the fitted value by more than 5 mpg.

Hover to identify a car. Drag to zoom, then double-click to reset.

library(ggplot2)
cars <- datasets::mtcars
fit <- lm(mpg ~ wt, data = cars)

ggplot(cars, aes(wt, mpg)) +
  geom_point(color = "#007C83", size = 3, alpha = 0.85) +
  geom_smooth(method = "lm", formula = y ~ x,
              color = "#8C1D40", fill = "#8C1D40", alpha = 0.13) +
  labs(x = "Weight (1,000 lb)", y = "Fuel economy (mpg)")

lm() estimates the coefficients. geom_smooth() draws the same fitted line and its confidence band.

• The model assumes a linear conditional mean and independent errors with constant variance.
• Exact small-sample t intervals also assume normally distributed errors. The residual plot alone cannot verify all assumptions.
• Weight may track other features, such as horsepower. This analysis does not establish that weight alone causes the mpg difference.
• Predictions are most defensible within the observed weight range: 1.51 to 5.42 thousand lb.

The fitted slope describes a strong negative association between weight and fuel economy in these 32 cars.

At 3,000 lb, the fitted mean is about 21.25 mpg, but the interval for an individual car is much wider.

The residual pattern makes a useful next question: would a curved relationship or an additional predictor improve the model?

Data: R datasets: mtcars. Methods: R’s stats::lm and stats::predict.lm. Plots: ggplot2 and Plotly. All numerical results are computed from the included R code.