Simple linear regression is one of the most common statistical tools.

It is used to:

• Describe relationships between variables
• Make predictions
• Quantify trends
• Evaluate strength of association

Example question:

Can vehicle weight predict fuel efficiency?

We will use the built-in R dataset mtcars.

The dataset contains information on 32 automobiles and includes measurements such as fuel efficiency, weight, horsepower, and number of cylinders.

Variables:

• mpg = Miles per gallon
• wt = Vehicle weight (1000 lbs)
• hp = Horsepower
• cyl = Number of cylinders

Our goal:

Predict mpg from wt

A simple linear regression model is:

\[ Y = \beta_0 + \beta_1X + \varepsilon \]

where

• \(Y\) = response variable
• \(X\) = predictor variable
• \(\beta_0\) = intercept
• \(\beta_1\) = slope
• \(\varepsilon\) = random error

For this example:

\[ mpg = \beta_0 + \beta_1(wt)+\varepsilon \]

Regression chooses coefficients that minimize:

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

where

\[ \hat y_i=\beta_0+\beta_1x_i \]

The method is called:

Ordinary Least Squares (OLS)

The fitted line minimizes the total squared residual error.

library(ggplot2)

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

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

Key findings from the regression model:

• Estimated slope = -5.34
• R² = 0.753
• p-value < 0.001
• Vehicle weight is a significant predictor of fuel efficiency
• Heavier vehicles tend to have lower MPG

Interpretation:

For every additional 1000 pounds of vehicle weight, fuel efficiency decreases by approximately 5.3 MPG.

Simple linear regression is a powerful statistical tool that:

• Quantifies relationships between variables
• Helps make predictions
• Provides interpretable results
• Is widely used in business, science, engineering, and data analytics

In this example, vehicle weight was a strong predictor of fuel efficiency.

Thank you!