• Simple linear regression studies the relationship between one predictor and one numerical outcome.
• It fits a straight line through observed data.
• We can use the line to describe a trend and make predictions.

Our question: Do heavier cars tend to get fewer miles per gallon (MPG)?

We will use the built-in R dataset mtcars, which contains measurements of 32 cars.

• Predictor (x): wt, car weight in thousands of pounds
• Outcome (y): mpg, fuel efficiency in miles per gallon
• Each dot on a graph represents one car.

This is an observational dataset, so a relationship does not automatically prove causation.

The dots generally slope downward: heavier cars tend to have lower MPG.

The general equation for a simple linear regression line is

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

• \(\hat{y}\) is the predicted outcome.
• \(b_0\) is the intercept, or predicted outcome when \(x=0\).
• \(b_1\) is the slope, or predicted change in the outcome for a one-unit increase in \(x\).

For our data, \(x\) is weight and \(\hat{y}\) is predicted MPG.

The line summarizes the average trend; the shaded band shows uncertainty about the mean fitted line.

Our estimated equation is

\[\widehat{MPG} = 37.29 - 5.34 \times Weight\]

• Slope: About 5.34 fewer MPG for each additional 1,000 pounds of car weight.
• Intercept: 37.29 MPG at zero weight. This is not a realistic car, so the intercept is not practically useful here.
• Prediction: A 3,000-pound car has a predicted fuel efficiency of about 21.3 MPG.

A residual is the difference between an observed value and its prediction:

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

The least-squares method chooses the line that minimizes the sum of squared residuals:

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

A positive residual means the observed MPG is higher than predicted. A negative residual means it is lower.

Hover over a point to see that car’s weight and MPG. You can also zoom into the chart.

The following R code fits the regression model, shows its coefficients, and predicts MPG for a 3,000-pound car.

model <- lm(mpg ~ wt, data = mtcars)
coef(model)

## (Intercept)          wt 
##   37.285126   -5.344472

predict(model, newdata = data.frame(wt = 3))

##        1 
## 21.25171

lm() fits a linear model; predict() uses the fitted model to estimate an outcome.

The coefficient of determination, \(R^2\), describes the share of variation in the outcome explained by the model.

\[R^2 = 1 - \frac{SSE}{SST}\]

For our regression, $R^2 = 0.753.

That means weight accounts for about 75.3% of the observed variation in MPG in this sample. Other factors also influence fuel efficiency.

• Simple linear regression helps describe and predict a numerical outcome using one predictor.
• In these 32 cars, greater weight is associated with lower MPG.
• The fitted slope gives an interpretable change in MPG for each additional 1,000 pounds.
• Predictions are estimates, not guarantees; this association alone does not establish cause and effect.

Data source: R built-in mtcars dataset (?mtcars).