Inference in Simple Linear Regression

The previous lesson produced numbers — $\hat\beta_0=0.146$, $\hat\beta_1=0.818$ for a hedge ratio — but numbers alone cannot answer the question a risk committee actually asks: is that slope reliably different from zero, or could five noisy days have produced this by chance? This note adds the one assumption needed to answer that — normality of the errors $\varepsilon_i$ — and derives the exact sampling distributions of $\hat\beta_0$ and $\hat\beta_1$. The payoff is threefold: a $t$-test for $H_0:\beta_1=0$, exact confidence intervals for the slope and intercept using the pivotal-quantity method already built, and a sharp distinction — routinely blurred in practice — between a confidence interval for the mean response $\mathbb{E}[Y\mid x]$ and a prediction interval for a single new observation at $x$.

Learning objectives. By the end of this note you should be able to: