Kenneth ChengMathematical modeling
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LEARN · EXPLAIN · REVISE

Linear regression and applications

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

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

Revisit first: Correlation of paired data

TOPIC 01

Linear regression and applications

Fit a least-squares line and distinguish prediction, residual and extrapolation.

What you will be able to explain

  • Fit a least-squares line and distinguish prediction, residual and extrapolation.
  • Justify the method and check the conditions in a new situation.

Defining relation

Fit a least-squares line and distinguish prediction, residual and extrapolation.

y^=a+bx;b=Sxy/Sxx;a=yˉ−bxˉ\hat y=a+bx;\quad b=S_{xy}/S_{xx};\quad a=\bar y-b\bar x

Conditions

Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Must the fitted line pass through every point?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Synthetic data: slope=2, intercept=1, r=1. Correlation alone does not establish causation.

Synthetic data: slope=2, intercept=1, r=1. Correlation alone does not establish causation.

Explain: Calculate two valid cases and explain the change using the defining relation.

Transfer: Compare residuals and the centroid property of a least-squares line.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find the fitted slope.

Sxx=2,Sxy=4S_{xx}=2,\quad S_{xy}=4
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
b=Sxy/Sxxb=S_{xy}/S_{xx}
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    b=4/2=2b=4/2=2

  3. The denominator is the variation in the predictor x.

The requested value is 2.

Checks and common pitfalls: The denominator is the variation in the predictor x.

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

02 / Standard#Worked example

For x=−1,0,1 and y=−t,0,t, find the least-squares slope.

t=3t=3
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
xˉ=yˉ=0\bar x=\bar y=0
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    Sxx=2,Sxy=6S_{xx}=2,\quad S_{xy}=6

  3. Apply the stated relation and retain its conditions.

    b=3b=3

  4. Preserve the paired order when calculating products.

The requested value is 3.

Checks and common pitfalls: Preserve the paired order when calculating products.

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

03 / Transfer#Worked example

For x=−1,0,1 and y=t,0,t, find the fitted line and explain what it misses.

t=4t=4
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
Sxy=0,yˉ=2t/3S_{xy}=0,\quad \bar y=2t/3
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    b=0,a=8/3b=0,\quad a=8/3

  3. The zero slope misses a U-shaped relation, so inspect the scatter as well as the line.

The requested relation or conclusion is shown below.

y^=8/3\hat y=8/3

Checks and common pitfalls: The zero slope misses a U-shaped relation, so inspect the scatter as well as the line.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

04 / Foundation#Your turn

Find the fitted slope.

Sxx=2,Sxy=10S_{xx}=2,\quad S_{xy}=10
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
b=Sxy/Sxxb=S_{xy}/S_{xx}
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    b=10/2=5b=10/2=5

  3. The denominator is the variation in the predictor x.

The requested value is 5.

Checks and common pitfalls: The denominator is the variation in the predictor x.

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

05 / Foundation#Your turn

Find the intercept.

xˉ=2,yˉ=15,b=6\bar x=2,\quad\bar y=15,\quad b=6
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
a=yˉ−bxˉa=\bar y-b\bar x
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    a=15−2(6)=3a=15-2(6)=3

  3. The fitted line passes through the data centroid.

The requested value is 3.

Checks and common pitfalls: The fitted line passes through the data centroid.

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

06 / Foundation#Your turn

Use the fitted line to predict y at x=2.

y^=3+7x\hat y=3+7x
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
y^(2)=3+2t\hat y(2)=3+2t
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    y^=17\hat y=17

  3. This is a fitted prediction rather than the observed response.

The requested value is 17.

Checks and common pitfalls: This is a fitted prediction rather than the observed response.

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

07 / Foundation#Your turn

An observation at x=2 has y=2t+5. Find its residual under ŷ=3+tx.

t=8t=8
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
e=y−y^e=y-\hat y
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    e=21−(19)=2e=21-(19)=2

  3. A positive residual means the observation is above the fitted line.

The requested value is 2.

Checks and common pitfalls: A positive residual means the observation is above the fitted line.

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

08 / Standard#Your turn

An observation at x=2 has y=2t+5. Find its residual under ŷ=3+tx.

t=9t=9
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
e=y−y^e=y-\hat y
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    e=23−(21)=2e=23-(21)=2

  3. A positive residual means the observation is above the fitted line.

The requested value is 2.

Checks and common pitfalls: A positive residual means the observation is above the fitted line.

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

09 / Standard#Your turn

For x=−1,0,1 and y=−t,0,t, find the least-squares slope.

t=10t=10
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
xˉ=yˉ=0\bar x=\bar y=0
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    Sxx=2,Sxy=20S_{xx}=2,\quad S_{xy}=20

  3. Apply the stated relation and retain its conditions.

    b=10b=10

  4. Preserve the paired order when calculating products.

The requested value is 10.

Checks and common pitfalls: Preserve the paired order when calculating products.

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

10 / Standard#Your turn

Data were observed only for 0≤x≤t. Assess a prediction at x=10t.

t=11t=11
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Compare the target x with the observed interval.
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    110>11110>11

  3. A good fit inside the sample range does not verify a far-out extrapolation.

The requested relation or conclusion is shown below.

extrapolation requires additional justification\text{extrapolation requires additional justification}

Checks and common pitfalls: A good fit inside the sample range does not verify a far-out extrapolation.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

11 / Standard#Your turn

For x=−1,0,1 and y=t,0,t, find the fitted line and explain what it misses.

t=12t=12
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
Sxy=0,yˉ=2t/3S_{xy}=0,\quad \bar y=2t/3
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    b=0,a=24/3b=0,\quad a=24/3

  3. The zero slope misses a U-shaped relation, so inspect the scatter as well as the line.

The requested relation or conclusion is shown below.

y^=24/3\hat y=24/3

Checks and common pitfalls: The zero slope misses a U-shaped relation, so inspect the scatter as well as the line.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

12 / Transfer#Your turn

Data were observed only for 0≤x≤t. Assess a prediction at x=10t.

t=13t=13
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Compare the target x with the observed interval.
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    130>13130>13

  3. A good fit inside the sample range does not verify a far-out extrapolation.

The requested relation or conclusion is shown below.

extrapolation requires additional justification\text{extrapolation requires additional justification}

Checks and common pitfalls: A good fit inside the sample range does not verify a far-out extrapolation.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

13 / Transfer#Your turn

For x=−1,0,1 and y=t,0,t, find the fitted line and explain what it misses.

t=14t=14
  • Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
Sxy=0,yˉ=2t/3S_{xy}=0,\quad \bar y=2t/3
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    b=0,a=28/3b=0,\quad a=28/3

  3. The zero slope misses a U-shaped relation, so inspect the scatter as well as the line.

The requested relation or conclusion is shown below.

y^=28/3\hat y=28/3

Checks and common pitfalls: The zero slope misses a U-shaped relation, so inspect the scatter as well as the line.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • Fit a least-squares line and distinguish prediction, residual and extrapolation.
    • Which condition is essential in linear regression and applications?
    • Must the fitted line pass through every point?

    Board plan

    • Defining relation: Fit a least-squares line and distinguish prediction, residual and extrapolation.
      y^=a+bx;b=Sxy/Sxx;a=yˉ−bxˉ\hat y=a+bx;\quad b=S_{xy}/S_{xx};\quad a=\bar y-b\bar x
    • Conditions: Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.

    Anticipated thinking

    • Residual means observed minus predicted, not the reverse.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗