Simple Linear Regression: mpg ~ wt

Predicting miles-per-gallon (mpg) from car weight (wt) using mtcars.

• Understand the simple linear regression model and OLS estimation.
  
• Fit the model in R and interpret output.
  
• Produce diagnostic and explanatory plots using ggplot2.
  
• Create an interactive 3D visualization with plotly.
  
• Include LaTeX math and R code.

The simple linear regression model:

\[ Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i,\quad \varepsilon_i\overset{iid}{\sim}N(0,\sigma^2) \]

Assumptions: linearity, independence, homoscedasticity, normal errors.

Ordinary Least Squares estimates:

\[ \hat{\beta}_1 = \frac{\sum_i (X_i-\bar X)(Y_i-\bar Y)}{\sum_i (X_i-\bar X)^2},\qquad \hat{\beta}_0 = \bar Y - \hat{\beta}_1\bar X. \]

Standard error of \(\hat{\beta}_1\):

\[ \mathrm{Var}(\hat{\beta}_1) = \frac{\sigma^2}{\sum_i (X_i-\bar X)^2} \quad\text{and}\quad \hat\sigma^2 = \frac{1}{n-2}\sum_i \hat\varepsilon_i^2. \]

data(mtcars)
knitr::kable(head(mtcars, 8))

# Fit the simple linear regression
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

This slide displays the R code and the model summary (estimates, t-stats, R-squared).

# Fit multiple regression and create a prediction grid for a plane
mod2 <- lm(mpg ~ wt + hp, data = mtcars)
# grid for surface
grid <- expand.grid(
  wt = seq(min(mtcars$wt), max(mtcars$wt), length.out = 25),
  hp = seq(min(mtcars$hp), max(mtcars$hp), length.out = 25)
)
grid$mpg_pred <- predict(mod2, newdata = grid)
# (The following creates a surface in plotly when viewed in the browser)

• Slope estimate: how many MPG units change per 1000 lbs change in weight (see coef(model)).
  
• R-squared: fraction of variance explained by weight.
  
• Diagnostics: check residuals for patterns; histogram for approximate normality.

• James et al., An Introduction to Statistical Learning (ISLR).
  
• R lm documentation and ggplot2 / plotly help pages.

*End of presentation