Simple linear regression uses a straight line to describe the relationship between two numerical variables.

Example: Does car weight help predict the fuel economy?

Must use R’s built-in mtcars dataset, which contains 32 cars.

• x: Weight in thousands of pounds (wt).
• y: Miles per gallon (mpg).

\[Y = \beta_0 + \beta_1x + \varepsilon\]

• \(\beta_0\) is the intercept.
• \(\beta_1\) is the slope: the change in mean \(Y\) when \(x\) increases by 1.
• \(\varepsilon\) represents unexplained variation.

The estimated line is

\[\widehat y = b_0 + b_1x.\]

R’s lm() function estimates the intercept and slope.

Heavier cars usually have lower fuel economy in this dataset.

An extra 1,000 lb is associated with about 5.34 fewer mpg. This association does not prove causation.

For a car weighing 3,000 lb, use \(x = 3\).

The fitted equation, with rounded coefficients, is

\[\widehat y = 37.285 - 5.344x.\]

Substitute \(x = 3\):

\[\widehat y = 37.285 - 5.344(3) \approx 21.25.\]

The predicted fuel economy is about 21.25 mpg. An individual car’s actual mpg can differ.

Hover over a point to see its values. Drag to zoom in.

library(ggplot2)
cars <- mtcars

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

ggplot(cars, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm", formula = y ~ x,
              se = FALSE) +
  labs(x = "Weight (1,000 lb)",
       y = "Miles per gallon")

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