| English | 의미 |
|---|---|
| least-squares regression/liːst skweəz rɪˈɡreʃn/ | 최소제곱 회귀 |
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.
적합된 기울기는 샘플마다 달라진다
각 샘플의 최소제곱 선은 약간의 다른 기울기 b를 가진다.
샘플의 기울기 b는 ...이다
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b는 불확실성을 가지고 미지 Population 기울기 β를 추정한다.
샘플 기울기 b의 Sampling 분포는 진짜 기울기 β를 중심으로 한다.
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b는 β에 대해 편향되지 않은 unbiased estimator이다.
기울기에 대한 추론은 어떤 분포를 사용하여 모델링하는가?
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평균과 같이, 기울기 추론은 t-분포를 사용한다.
회귀 추론은 Unit 2의 최소제곱 모델 ŷ = a + bx를 기반으로 한다.
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Unit 2의 기울기 주변에 t-구간과 검정을 추가한다.
b의 Sampling 분포의 퍼짐은 ___ 의 표준 오차이다 (단어 하나).
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SE_b는 기울기의 표준 오차이다.