LEARN · EXPLAIN · REVISE

A survey links tutoring hours and exam scores. Does the regression establish tutoring as the cause?

Read the idea, work independently, then explain what changed.

Return to the lesson / paper ↗

高三選擇性必修第三册(A版).pdf · 8.2 · PDF 110 / printed page 105

TOPIC 01

Paired data, fitted lines and responsible prediction

Calculate a least-squares line from paired centered data and interpret residuals and units.

What you will be able to explain

Calculate a least-squares line from paired centered data and interpret residuals and units.
Distinguish linear association, nonlinear dependence, extrapolation and causal claims.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Foundation#Your turn

A survey links tutoring hours and exam scores. Does the regression establish tutoring as the cause?

\text{observational survey}

Keep the observed pairs; unique slope needs Sxx>0, correlation also Syy>0, and causal claims need a suitable design.

Skills and prerequisite lessons

Scope: School extension · This task develops the school textbook skill beyond the explicitly listed JAE syllabus topics.

Working and explanation

BUILD THE REASONING

Hint 1

Separate an observational fit from a causal design.

Hint 2

Start from this intermediate relation.

\text{association}\not\Rightarrow\text{causation}

Worked solution

Separate an observational fit from a causal design.
Establish and simplify this mathematical relation.
$\text{association}\not\Rightarrow\text{causation}$
Establish and simplify this mathematical relation.
$\text{selection and confounding require investigation}$
A precise slope cannot repair a design that does not identify causation.

No; prior attainment and selection can confound the association.

Checks and common pitfalls: A precise slope cannot repair a design that does not identify causation.

Reasoning checklist · self / teacher assessment

Separate an observational fit from a causal design.
Give the intermediate relations and explain the conclusion with its restrictions.
A precise slope cannot repair a design that does not identify causation.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Predict	Predict how a raised final y-value changes the fitted slope.
8	Build the idea	Sample response: keep pairs and use centered Sxy/Sxx; do not sort each column separately.
10	Guided comparison	Sample response: observed 8 minus predicted 7 gives residual 1.
12	Independent practice and discussion	Discuss unit changes, pairing reversal and choice of squared versus absolute loss.
5	Exit and next step	Exit: a fitted line is not causal evidence, and extrapolation to x=100 needs separate justification.

Question sequence

Predict how a raised final y-value changes the fitted slope.
Sample response: observed 8 minus predicted 7 gives residual 1.

Board development

Fit a line to pairs, not independent lists
$b=S_{xy}/S_{xx};\ a=\bar y-b\bar x;\ e=y-\hat y$

Expected correct and incorrect responses

Correct sample: 2
Anticipated wrong response: −2 — Subtracts observed from predicted response.
Anticipated wrong response: Yes, all variables are independent. — Interprets zero linear correlation as independence.
Reversing residual subtraction.
Equating correlation with independence or causation.

Follow-up questions

Follow-up: Subtract predicted from observed. Then explain why −2 fails.
Follow-up: Consider a symmetric quadratic scatter. Then explain why Yes, all variables are independent. fails.
Explain why an overall response shift changes the intercept but leaves residuals and slope unchanged.

Remediation

Sample response: keep pairs and use centered Sxy/Sxx; do not sort each column separately.
Return to the first example, identify the missing condition, then solve one fresh task without viewing its solution.

Exit check

Exit: a fitted line is not causal evidence, and extrapolation to x=100 needs separate justification.

Assessment checklist and sample responses

Representation and conditions
Keep each pair and check Sxx=2>0 before dividing.
Connected mathematical reasoning
For (1,1),(2,3),(3,2), Sxy=1 and Sxx=2, so slope is 1/2.
Interpretation and boundary check
A symmetric y=x² scatter can have r=0, so zero correlation does not imply independence.