Do Those Points Align?
Introduced| English |
|---|
| least-squares regression/liːst skweəz rɪˈɡreʃn/ |
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.
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."
The slope $b$ from one sample is an estimate, not the truth $\beta$. A different random sample would give a different $b$. Inference for slopes is exactly the same logic as for means — a $t$-based interval and test around a sample statistic — applied to the regression slope, with its own standard error $SE_b$.
Fitting hours-studied vs. score for one class gives slope $b = 4.2$.
- A different class would give a different $b$ (maybe $3.8$ or $4.6$) — sampling variability.
- The true slope $\beta$ for all students is unknown; $b = 4.2$ estimates it.
- Unit 9 builds an interval and a test for that unknown $\beta$.
The slope $b$ of a fitted line varies from sample to sample; its sampling distribution is centered at the true slope $\beta$ with spread $SE_b$, modeled by a $t$-distribution. Inference for slopes quantifies the uncertainty about the true linear relationship, building directly on the least-squares regression model.
A fitted slope varies by sample
Each sample's least-squares line has a slightly different slope b.
The slope b from a sample is...
b estimates the unknown population slope β, with uncertainty.
The sampling distribution of the sample slope b is centered at the true slope β.
b is unbiased for β.
Inference for slopes is modeled using which distribution?
Like means, slope inference uses a t-distribution.
Regression inference builds on the least-squares model ŷ = a + bx from Unit 2.
It adds a t-interval and test around the Unit 2 slope.
The spread of the sampling distribution of b is the standard error of the ___ (one word).
SE_b is the standard error of the slope.