18. Inference for Linear Regression

What Is the Line Estimating?

Topic 17 fit a line to a sample: \hat{y} = b_0 + b_1 x. Behind that sample sits a population, where every individual follows

y = \beta_0 + \beta_1 x + \varepsilon

\beta_0 and \beta_1 are the population intercept and slope — fixed, unknown, never observed. \varepsilon is how far one individual falls above or below that line.

b_0 and b_1 estimate \beta_0 and \beta_1. A different sample gives different estimates — so what can one sample’s b_1 tell us about \beta_1? That question is inference for the slope.

What Does “No Relationship” Look Like?

A different population — this time x tells you nothing about y:

Try it — in y = \beta_0 + \beta_1 x + \varepsilon, what must \beta_1 be?

\beta_1 = 0, and the model collapses to y = \beta_0 + \varepsilon. x is gone from the equation. Every x gives the same predicted y, so the population line is flat. Here \beta_0 = 50.

Do Samples Come Out Flat?

Draw a sample of 20 from that population — the one where \beta_1 = 0.

Same population every time, with \beta_1 = 0. But the sample slope b_1 never comes out zero.
So the question: is b_1 far enough from zero to rule out \beta_1 = 0?

Hypotheses for the Slope

Try it — the research question is whether food rating is related to price. Write H_0 and H_A in terms of \beta_1.

H_0: \beta_1 = 0 — the true slope is zero; a restaurant’s food rating tells you nothing about its price, linearly.
H_A: \beta_1 \neq 0 — there is a linear relationship; food rating and price move together.

Two-sided is the default. One-sided versions (\beta_1 > 0 or \beta_1 < 0) exist when the research question commits to a direction.

Test Statistic and Degrees of Freedom

Same T-statistic shape we’ve used since topic 11:

T = \frac{\text{statistic} - \text{null value}}{SE} = \frac{b_1 - 0}{SE_{b_1}} = \frac{b_1}{SE_{b_1}}

Software reports b_1 and SE_{b_1} — you can always verify T by hand.

Reference distribution: under H_0, T follows a t-distribution with

df = n - 2

Why n - 2? Fitting the line uses up two degrees of freedom — one for the intercept b_0, one for the slope b_1.

Conditions for Inference

Four conditions for the t-distribution to be a valid reference for T: