Can vehicle weight help predict miles per gallon (MPG)?

• Visualize the relationship between weight and fuel economy.
• Explain the least-squares regression model and its coefficients.
• Calculate a prediction and evaluate limitations of the model.
• Use ggplot2 and interactive Plotly to explore the data.

• Response (Y): mpg = miles traveled per gallon of fuel.
• Predictor (X): wt = car weight in thousands of pounds.
• Additional variable: hp = horsepower, explored in the 3D plot.

Simple linear regression models the average MPG at a given weight:

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

\[\widehat{Y}_i = \widehat{\beta}_0 + \widehat{\beta}_1 X_i\]

• \(Y_i\): observed MPG; \(X_i\): weight in units of 1,000 lb.
• \(\beta_0\): intercept; \(\beta_1\): change in mean MPG for an additional 1,000 lb.
• \(\varepsilon_i\): unexplained difference between observation and population mean.

Choose the intercept and slope that minimize squared vertical errors:

\[\operatorname{SSE}(b_0,b_1)=\sum_{i=1}^{n}[y_i-(b_0+b_1x_i)]^2\]

\[\widehat{\beta}_1=\frac{\sum_i(x_i-\bar{x})(y_i-\bar{y})}{\sum_i(x_i-\bar{x})^2}\]

\[\widehat{\beta}_0=\bar{y}-\widehat{\beta}_1\bar{x}\]

Intuition: Find the line with the smallest overall sum of squared misses.

This R code generates the regression results and a prediction:

# Built-in data; no CSV files required.
fit <- lm(mpg ~ wt, data = mtcars)
summary(fit)       # Coefficients, R-squared, tests
predict(fit, newdata = data.frame(wt = 3))

\[\widehat{\mathrm{MPG}} = 37.29 - 5.34\,(\mathrm{Weight\ in\ 1{,}000\ lb})\]

• Slope: -5.34 MPG per additional 1,000 lb, on average.
• Intercept: 37.29 MPG at weight 0; outside the data’s meaningful range.
• \(R^2\): 0.753, meaning about 75.3% of MPG variation within this sample is accounted for by the fitted linear relationship.
• Interpretation: Association does not by itself prove causation.

Since weight is measured in thousands of pounds, use \(x=3\):

\[\widehat{\mathrm{MPG}}=\widehat{\beta}_0+\widehat{\beta}_1(3)\]

\[\widehat{\mathrm{MPG}}=37.29 - 5.34(3)\]

\[\boxed{\widehat{\mathrm{MPG}}\approx 21.25}\]

A predicted value is not a guaranteed MPG for an individual vehicle.

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

• The fitted model’s \(R^2\) is 0.753.
• This measures how much of the sample’s observed variability is explained by the fitted relationship.
• A larger \(R^2\) alone does not validate assumptions or show that predictions generalize.

This interactive chart shows MPG versus weight. Hover over any point to see the vehicle’s name, MPG, weight, and horsepower; color also represents horsepower.

• Linearity: The conditional mean relationship should be approximately linear.
• Independence: Observations/errors should be independent for standard inference.
• Constant error spread: Residual variation should be reasonably consistent.
• Normal errors: Relevant primarily to small-sample inference, not fitting the line.
• Scope: Data from older cars; modern vehicles may follow different patterns.

• Among these 32 cars, heavier weight is associated with lower MPG.
• Linear regression summarizes that relationship with an interpretable equation.
• For a 3,000-lb car, the estimated MPG is 21.25.
• The residual plot and interactive Plotly chart reveal why data exploration matters.
• Bottom line: Useful predictions require thoughtful interpretation and awareness of limitations.

Data source: R’s built-in mtcars dataset (?mtcars).
Tools: R, ggplot2, Plotly.js, jsonlite, and knitr. All plots and calculations are generated in this file.