• This presentation will explore Simple Linear Regression.
• We will use the built-in mtcars dataset in R.
• The goal is to model the relationship between a car’s weight (wt) and its fuel efficiency (mpg).

The core of simple linear regression is a linear equation that models the relationship between a dependent variable (Y) and an independent variable (X).

The formula is: \[ Y = \beta_0 + \beta_1 X + \epsilon \]

• \(Y\) is the dependent variable (mpg)
• \(X\) is the independent variable (wt)
• \(\beta_0\) is the y-intercept
• \(\beta_1\) is the slope
• \(\epsilon\) represents the error term

First, let’s visualize the relationship between weight and MPG with a scatter plot. This helps us see if a linear relationship is a reasonable assumption.

We use the lm() function in R to create the linear model. The formula mpg ~ wt tells R to model mpg as a function of wt.

library(broom)
library(knitr)

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

kable(tidy(model), digits = 3)

Now we can add the regression line calculated by our mode l to the scatter plot. This line represents the predicted mpg for any given wt.

From the model summary, we get our coefficients:

• Intercept (\(\beta_0\)): 37.285
• Slope (\(\beta_1\)): -5.344

This means that for every 1000 lbs increase in weight, the car’s fuel efficiency is predicted to decrease by 5.34 mpg.

• This presentation successfully demonstrated a simple linear regression analysis.
• We found a clear, negative relationship between a car’s weight and its fuel efficiency.