Does a heavier car tend to have worse fuel economy?

• We measure fuel economy in miles per gallon (mpg).
• Vehicle weight is measured in thousands of pounds (wt).
• Our goal is to describe the relationship, quantify its strength, and make an informed prediction.

The built-in R dataset mtcars contains measurements for 32 automobile models from a 1974 automotive magazine dataset.

• wt: weight in units of 1,000 pounds
• mpg: fuel economy in miles per gallon
• cyl: number of cylinders; useful context when comparing vehicles

In this sample, weights range from 1.51 to 5.42 thousand pounds.

Source: R built-in mtcars dataset; documentation: ?mtcars.

Interpretation: Heavier vehicles generally have lower mpg, suggesting a negative association.

The Pearson correlation coefficient measures the direction and strength of a linear relationship.

\[ r = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})} {\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2}\sqrt{\sum_{i=1}^{n}(y_i-\bar{y})^2}} \]

For these 32 vehicles, \(r = -0.868\).

• Negative sign: higher weight tends to go with lower mpg.
• Magnitude close to 1: the linear association is strong in this sample.

We model average mpg as a straight-line function of weight:

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

The fitted line is:

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

• Intercept: a mathematical anchor, not a realistic prediction for a zero-weight car.
• Slope: each additional 1,000 lb is associated with about 5.34 fewer mpg on average in the fitted model.

Interpretation: The downward line shows the fitted average trend. The shaded region is a 95% confidence band for mean mpg, not an individual-car prediction interval.

\[ R^2 = 1 - \frac{\sum_i (y_i-\hat y_i)^2} {\sum_i (y_i-\bar y)^2} \]

Our regression has \(R^2 = 0.753\), or about 75.3%.

The remaining variation may involve engine design, aerodynamics, gearing, and other factors not included here.

Hover over a point to identify the car and its weight, mpg, and cylinder count. Drag to zoom; double-click to reset.

Here is the essential code used to calculate the relationship and draw the regression line.

# mtcars is included with R
library(ggplot2)
data(mtcars)

fit <- lm(mpg ~ wt, data = mtcars)
cor(mtcars$wt, mtcars$mpg)
summary(fit)$r.squared

ggplot(mtcars, aes(wt, mpg)) +
  geom_point() +
  geom_smooth(method = "lm", se = TRUE)

R functions: lm() fits the linear model; cor() measures linear association; ggplot() and geom_smooth() visualize the data and fitted trend.

Using the fitted line, consider two hypothetical vehicle weights within the observed range:

The heavier example is predicted to achieve about 5.3 fewer mpg.

These are model predictions for average mpg, not guaranteed outcomes for particular cars.

What supports our conclusion?

• Scatterplot and regression line both show a downward pattern.
• The correlation is -0.868, indicating a strong negative linear relationship.
• The model’s \(R^2\) is 0.753.

What should we be careful about?

• Only 32 older automobile models are included.
• Engine type, cylinders, and design differences could confound the relationship.
• The fitted line should not be extrapolated far beyond the observed weights.
• Association does not establish causation.

Question: Does vehicle weight help explain fuel economy?

Answer: Yes, in mtcars, heavier vehicles tend to have lower mpg. Simple linear regression makes this pattern interpretable and quantifiable.

What we learned:

1. Statistics turns an apparent pattern into a measurable relationship.
2. Mathematics defines correlation, a regression line, and explained variation.
3. R, ggplot2, and Plotly communicate the evidence through static and interactive visuals.

Data: built-in R mtcars dataset. Methods and visualizations created with base R, ggplot2, and Plotly.