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All observed x-values equal 2. Can Sxy/Sxx determine a unique fitted slope?

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 / Foundation#Your turn

All observed x-values equal 2. Can Sxy/Sxx determine a unique fitted slope?

x_i=2\text{ for all }i

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

Check variation in the predictor before dividing.

Hint 2

Start from this intermediate relation.

S_{xx}=\sum_i(x_i-2)^2=0

Worked solution

Check variation in the predictor before dividing.
Establish and simplify this mathematical relation.
$S_{xx}=\sum_i(x_i-2)^2=0$
Establish and simplify this mathematical relation.
$a+2b=\bar y$
The average response is fitted, but slope is not identifiable.

No; Sxx=0, and infinitely many intercept/slope pairs give the same fitted constant on the data.

Checks and common pitfalls: The average response is fitted, but slope is not identifiable.

Reasoning checklist · self / teacher assessment

Check variation in the predictor before dividing.
Give the intermediate relations and explain the conclusion with its restrictions.
The average response is fitted, but slope is not identifiable.

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.