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Observed y=8 and predicted ŷ=6. What is 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.

01 / Foundation#Your turn

Observed y=8 and predicted ŷ=6. What is the residual?

y=8; y^=6y=8;\ \hat y=6
  • 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.

  1. −2
  2. 14
  3. 2
  4. 6
Working and explanation
BUILD THE REASONING
Hint 1
Subtract predicted from observed.
Hint 2
Start from this intermediate relation.
e=y−y^e=y-\hat y
Worked solution
  1. Subtract predicted from observed.
  2. Establish and simplify this mathematical relation.
    e=y−y^e=y-\hat y
  3. Establish and simplify this mathematical relation.
    e=2e=2
  4. The sign records which side of the fitted line the observation lies on.
2
Checks and common pitfalls: The sign records which side of the fitted line the observation lies on.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5PredictPredict how a raised final y-value changes the fitted slope.
8Build the ideaSample response: keep pairs and use centered Sxy/Sxx; do not sort each column separately.
10Guided comparisonSample response: observed 8 minus predicted 7 gives residual 1.
12Independent practice and discussionDiscuss unit changes, pairing reversal and choice of squared versus absolute loss.
5Exit and next stepExit: 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=Sxy/Sxx; a=yˉ−bxˉ; e=y−y^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.
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Curriculum and source notes ↗