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語彙

English	意味
least-squares regression/liːst skweəz rɪˈɡreʃn/	最小二乗回帰

English

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.

English

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.

English

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.

English

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."

English

Watch out

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 $SE_b$.

English

Example

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 $\beta$.

English

Key idea

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.

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