Simple Linear Regression models the relationship between two quantitative variables using a straight line. We will model MPG (miles per gallon) as a function of Weight using the built-in mtcars dataset.

\[ Y_i = \beta_0 + \beta_1 X_i + \epsilon_i, \quad \epsilon_i \sim \mathcal{N}(0, \sigma^2) \] - \(\beta_0\): intercept
- \(\beta_1\): slope (change in \(Y\) per one-unit change in \(X\))

library(ggplot2)
library(plotly)         # for interactive 3D plot
data(mtcars)
# Global chunk options to avoid overflowing slides
knitr::opts_chunk$set(
  fig.width = 6.5,      # wide but not too tall
  fig.height = 4.2,
  out.width = "100%",
  dpi = 96,
  message = FALSE,
  warning = FALSE
)

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

summary(model)

## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

coef(model)

## (Intercept)          wt 
##   37.285126   -5.344472

The fitted line is \[ \hat{Y} = \hat{\beta}_0 + \hat{\beta}_1 X. \] - If \(\hat{\beta}_1 < 0\): as weight increases, MPG decreases.
- \(R^2\) quantifies goodness-of-fit.

• Heavier cars tend to have lower fuel efficiency.
  
• Linear regression provides an interpretable baseline model.
  
• Requirements covered: ggplot2 (2 plots), plotly (1 interactive 3D), ≥2 math slides, code slides, 8+ total slides.