We often want to explain or predict a numeric outcome using one predictor.

Example application:
Predicting fuel efficiency (mpg) using vehicle weight (wt) from the built-in mtcars dataset.

• Response variable: mpg
• Predictor variable: wt (in 1000 lbs)

Simple linear regression assumes:

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

where: - \(Y_i\) is the response - \(X_i\) is the predictor - \(\varepsilon_i \sim N(0,\sigma^2)\)

• \(\beta_0\): expected mpg when weight is 0
  
• \(\beta_1\): expected change in mpg for a 1-unit increase in weight

Estimated regression line:

\[ \hat{Y} = \hat{\beta}_0 + \hat{\beta}_1 X \]

We test:

\[ H_0: \beta_1 = 0 \quad \text{vs} \quad H_A: \beta_1 \neq 0 \]

Test statistic:

\[ t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)}, \quad df = n - 2 \]

The p-value measures evidence against \(H_0\).

• Confidence interval: uncertainty about the mean mpg
  
• Prediction interval: uncertainty for a new car (wider)

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

ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm", se = TRUE) +
  labs(title = "mpg vs wt with linear fit")

• Simple linear regression models a relationship between two variables
• ggplot helps visualize the relationship and diagnostics
• p-values assess statistical significance
• Prediction intervals are wider than confidence intervals
• plotly enables interactive 3D exploration