What is linear regression?

• Linear regression models the relationship between an outcome and a predictor.
• Predictor: car weight (wt), in 1,000 pounds.
• Outcome: fuel efficiency (mpg), in miles per gallon.
• Question: do heavier cars tend to have lower gas mileage?

Our example: the mtcars dataset

• mtcars comes with R and contains information on 32 cars.
• We will use wt to predict mpg.
• The data show association, not proof that weight alone causes mileage differences.

The regression equation

We estimate a straight-line relationship:

\[\widehat{y} = b_0 + b_1 x\]

• \(\widehat{y}\) is the predicted MPG.
• \(x\) is the car’s weight (in thousands of pounds).
• \(b_0\) is the intercept; \(b_1\) is the slope.

The slope tells us how predicted MPG changes when weight increases by 1,000 pounds.

Scatterplot: weight vs. MPG

Fit the model in R

This code estimates the line using ordinary least squares.

fit <- lm(mpg ~ wt, data = mtcars)
round(coef(fit), 2)

## (Intercept)          wt 
##       37.29       -5.34

round(summary(fit)$r.squared, 3)

## [1] 0.753

The lm() function chooses the line that minimizes squared residuals.

A prediction with the fitted equation

From our sample, the fitted model is approximately:

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

For a car weighing 3,000 pounds (wt = 3):

\[\widehat{\mathrm{MPG}} = 37.29 - 5.34(3) \approx 21.27\]

That is a predicted 21.3 miles per gallon, not an exact outcome.

Residuals and squared error

A residual is observed MPG minus predicted MPG:

\[e_i = y_i - \widehat{y}_i\]

Ordinary least squares minimizes:

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

Small residuals mean the model fits those observations more closely.

Residual plot (second ggplot)

Patterns in residuals can suggest that a straight-line model misses part of the relationship.

Interactive scatterplot: weight vs. MPG (Plotly)

Move your mouse over each dot to see the car model, weight, and fuel efficiency. You can also zoom in on the plot.

How well does the line explain MPG?

• The fitted slope is -5.34 MPG per 1,000 pounds.
• The model’s \(R^2\) is 0.753.
• So weight explains about 75.3% of the sample variation in MPG through this linear model.
• Other factors, such as horsepower, may also be related to MPG.

Takeaways

• Simple linear regression summarizes a relationship using a fitted line.
• In these 32 cars, more weight is associated with lower MPG.
• Scatterplots help visualize the trend; residuals help assess fit.
• Predictions should be treated cautiously outside the weight range in the sample.

Data: built-in R dataset mtcars. Plots created with ggplot2 and Plotly.