Linear Regression

Modeling the relationship between independent ($X$) and dependent ($Y$) variables.

Simple Linear Model

$Y_j = \alpha + \beta X_j + \epsilon_j$ where errors $\epsilon_j \sim N(0, \sigma^2)$.

Least Squares

We find the line $y = a + bx$ that minimizes the Sum of Squared Errors (SSE).

Inference in Regression

We can test if the relationship is real (Test of Utility): $H_0: \beta = 0$ If rejected, $X$ provides significant predictive power for $Y$.