LEARN · EXPLAIN · REVISE

For fitted ŷ=1+2x and observation (3,8), find the residual.

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

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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 / Standard#Worked example

For fitted ŷ=1+2x and observation (3,8), find the residual.

\hat y=1+2x;\ (x,y)=(3,8)

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

Subtract predicted from observed response.

Hint 2

Start from this intermediate relation.

\hat y(3)=7

Worked solution

Subtract predicted from observed response.
Establish and simplify this mathematical relation.
$\hat y(3)=7$
Establish and simplify this mathematical relation.
$e=8-7=1$
Reversing the subtraction changes the residual sign.

The requested value is 1.

Checks and common pitfalls: Reversing the subtraction changes the residual sign.

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.