Simple linear regression examines the relationship between one predictor variable and one response variable.

For this example, we will study how vehicle weight affects fuel economy.

A simple linear regression model is:

\[ Y_i = \beta_0 + \beta_1X_i + \epsilon_i \]

Where:

• \(\beta_0\) = intercept
• \(\beta_1\) = slope
• \(\epsilon_i\) = random error

We will use the built-in R dataset mtcars.

Important variables:

• mpg = miles per gallon
• wt = vehicle weight in thousands of pounds
• hp = horsepower

##                    mpg    wt  hp
## Mazda RX4         21.0 2.620 110
## Mazda RX4 Wag     21.0 2.875 110
## Datsun 710        22.8 2.320  93
## Hornet 4 Drive    21.4 3.215 110
## Hornet Sportabout 18.7 3.440 175
## Valiant           18.1 3.460 105

## `geom_smooth()` using formula = 'y ~ x'

Heavier vehicles generally have lower fuel economy.

The estimated regression equation is:

\[ \hat{Y} = b_0 + b_1X \]

## (Intercept)          wt 
##   37.285126   -5.344472

The slope shows how MPG changes as vehicle weight increases.

The coefficient of determination is:

\[ R^2 = 1 - \frac{SS_{res}}{SS_{tot}} \]

## [1] 0.7528328

A higher \(R^2\) means the model explains more of the variation in MPG.

Residuals help determine whether the linear model is reasonable.

This interactive graph compares weight, horsepower, and fuel economy.

library(ggplot2)

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

ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")

Suppose a vehicle weighs 3,000 pounds.

Since wt is measured in thousands:

\[ X = 3 \]

##        1 
## 21.25171

The model predicts the expected MPG for a vehicle weighing 3,000 pounds.

• Linear regression studies relationships between quantitative variables.
• The slope measures how the response changes.
• \(R^2\) measures how well the model explains the data.
• Residuals help evaluate the model.
• Vehicle weight is strongly related to fuel economy.