Simple linear regression helps us understand how one numerical variable changes as another changes.

In this presentation, we study the relationship between:

• vehicle weight (wt)
• fuel efficiency in miles per gallon (mpg)

Using the mtcars dataset, we will see how regression helps describe patterns, make predictions, and test whether a relationship is statistically meaningful.

Our guiding question is:

Does a car’s weight help explain its fuel efficiency?

This is a useful regression example because both variables are quantitative, and the relationship appears approximately linear.

If heavier cars tend to have lower gas mileage, a regression model should help quantify that pattern.

Simple linear regression models the relationship between a response variable and one explanatory variable as

\[ Y_i = \beta_0 + \beta_1 x_i + \varepsilon_i \]

where:

• \(Y_i\) is the response
• \(x_i\) is the explanatory variable
• \(\beta_0\) is the intercept
• \(\beta_1\) is the slope
• \(\varepsilon_i\) is the random error term

In our example:

• \(Y_i = mpg\)
• \(x_i = wt\)

A common assumption is

\[ \varepsilon_i \sim N(0,\sigma^2) \]

The least-squares method chooses the line that minimizes squared prediction errors.

The slope estimate is

\[ \hat{\beta}_1 = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2} \]

Predicted values are computed as

\[ \hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i \]

To test whether weight is useful, we test

\[ H_0:\beta_1 = 0 \qquad \text{vs.} \qquad H_a:\beta_1 \ne 0 \]

We fit the model

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

using the mtcars dataset.

Important results from the fitted model:

• Estimated intercept: 37.29
• Estimated slope: -5.34
• \(R^2\): 0.753
• p-value for the slope: 1.29^{-10}

Interpretation:

For every additional 1000 pounds of vehicle weight, the model predicts that fuel efficiency decreases by about 5.34 miles per gallon on average.

This scatterplot shows a clear negative trend.

As vehicle weight increases, fuel efficiency tends to decline.

library(ggplot2)
library(plotly)

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

ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point(size = 3) +
  geom_smooth(method = "lm", se = TRUE) +
  labs(
    title = "Fuel Efficiency vs. Car Weight",
    x = "Weight (1000 lbs)",
    y = "Miles per Gallon"
  ) +
  theme_minimal()

Residuals are scattered around zero, which suggests the linear model is a reasonable first step.

Hovering over the points allows us to inspect individual observations more closely.

This analysis shows that simple linear regression can:

• describe the direction and strength of a relationship
• provide predictions
• test whether a variable is statistically significant

In the mtcars data, weight is a strong negative predictor of fuel efficiency.

Simple linear regression gives a useful first look at how two quantitative variables are related.

In this case, the conclusion is clear:

Heavier cars tend to get lower gas mileage, and the regression model captures that pattern effectively.

A natural next step would be multiple regression using additional variables such as horsepower or cylinders.