Do Those Points Align?

The slope wobbles too

• In Unit 2 we fit a least-squares line and read its slope $b$.
• But $b$ comes from a sample — a different sample gives a slightly different slope.
• So the slope varies from sample to sample, just like $\bar{x}$ or $\hat{p}$.
• Unit 9 does inference about the true slope $\beta$ behind the data.

The sampling distribution of b

• Collect the slope $b$ from every possible sample → its sampling distribution.
• It's centered at the true population slope $\beta$ (so $b$ is unbiased for $\beta$).
• Its spread is the standard error of the slope, $SE_b$.
• Under the right conditions, this distribution is modeled by a $t$-distribution.

A line of best fit runs through the middle of the scattered points

Inference about the relationship

• Inference for slopes asks: is there really a linear relationship, and how strong?
• The true slope $\beta$ is unknown; $b$ estimates it with uncertainty.
• A confidence interval gives a range for $\beta$; a test judges whether $\beta = 0$.
• Both quantify how sure we are about the line behind the scatter.

Built on Unit 2

• Everything rests on the least-squares regression 最小二乘回归 model $\hat{y} = a + bx$.
• $b$ is still the predicted change in $y$ per unit $x$ — now with error bars.
• We add a $t$-based interval and test around that slope.
• Regression inference is "Unit 2's slope, plus Unit 7's $t$-machinery."

Write from a slope that varies

• Many random samples are taken from one linear relationship.
• Each sample has its own least-squares slope.