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Paired data, fitted lines and responsible prediction

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

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.

Fit a line to pairs, not independent lists

A prediction summarizes a relationship but does not force every observation to lie on the line.

Least squares chooses the line minimizing squared vertical residuals and passing through the centroid when an intercept is fitted.

b=S_{xy}/S_{xx};\ a=\bar y-b\bar x;\ e=y-\hat y

Least squares chooses the line minimizing squared vertical residuals and passing through the centroid when an intercept is fitted.
$\partial SSE/\partial a=-2\sum_i(y_i-a-bx_i)=0$
Least squares chooses the line minimizing squared vertical residuals and passing through the centroid when an intercept is fitted.
$\partial SSE/\partial b=0\implies b=S_{xy}/S_{xx}$

When it applies

Sxx>0 for a unique slope; correlation also needs Syy>0.
Prediction beyond the observed range and causal interpretation need additional evidence.

Counterexample: The symmetric points of y=x² can have r=0 despite a deterministic nonlinear relation.

Paired table, scatterplot, residual segments, centroid and squared-error calculation.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Predict how changing the last paired response and shifting all responses affect slope, intercept and residuals.

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

Fitted ŷ=0+2x. Shifting every response changes the intercept only; changing the last pair also changes slope.

Numerical data for the diagram; printable
Input	Observed	Predicted
1	2	2
2	4	4
3	6	6
4	8	8
5	10	10

Explain: Read the scatter, fitted line and printable table of observed/predicted/residual values.

Transfer: Explain why an overall response shift changes the intercept but leaves residuals and slope unchanged.

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#Worked example

For pairs (1,1),(2,3),(3,2), find the least-squares slope.

(x,y)=(1,1),(2,3),(3,2)

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

Center both coordinates while preserving pairs.

Hint 2

Start from this intermediate relation.

\bar x=\bar y=2;\ S_{xx}=2;\ S_{xy}=1

Worked solution

Center both coordinates while preserving pairs.
Establish and simplify this mathematical relation.
$\bar x=\bar y=2;\ S_{xx}=2;\ S_{xy}=1$
Establish and simplify this mathematical relation.
$b=S_{xy}/S_{xx}=1/2$
A slope between selected endpoints need not be the fitted slope.

The requested value is 0.5.

Checks and common pitfalls: A slope between selected endpoints need not be the fitted slope.

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

02 / Standard#Worked example

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

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

First step provided — complete the reasoning:Subtract predicted from observed response.

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.

03 / Transfer#Worked example

A fitted slope is 2 when x is measured in hours. If x′=60x measures minutes, find the new slope. Explain what remains unchanged.

\hat y=a+2x;\ x\prime=60x

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

Substitute the new coordinate into the same fitted relation.

Hint 2

Start from this intermediate relation.

x=x\prime/60

Worked solution

Substitute the new coordinate into the same fitted relation.
Establish and simplify this mathematical relation.
$x=x\prime/60$
Establish and simplify this mathematical relation.
$\hat y=a+(1/30)x\prime$
The numerical slope changes with input units while paired predictions stay the same.

The requested value is 0.03333333333333333.

Checks and common pitfalls: The numerical slope changes with input units while paired predictions stay the same.

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

04 / Foundation#Your turn

A fitted slope is 3 and the sample means are x̄=2,ȳ=8. Find the intercept.

b=3;\ \bar x=2;\ \bar y=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

The least-squares line with intercept passes through the centroid.

Hint 2

Start from this intermediate relation.

a=\bar y-b\bar x

Worked solution

The least-squares line with intercept passes through the centroid.
Establish and simplify this mathematical relation.
$a=\bar y-b\bar x$
Establish and simplify this mathematical relation.
$a=8-3(2)=2$
The centroid property does not require the line to pass through every observation.

The requested value is 2.

Checks and common pitfalls: The centroid property does not require the line to pass through every observation.

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

05 / Foundation#Your turn

For ŷ=2+3x, predict y at x=4.

\hat y=2+3x;\ x=4

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

Use the fitted equation and distinguish prediction from observation.

Hint 2

Start from this intermediate relation.

\hat y(4)=2+3(4)

Worked solution

Use the fitted equation and distinguish prediction from observation.
Establish and simplify this mathematical relation.
$\hat y(4)=2+3(4)$
Establish and simplify this mathematical relation.
$\hat y(4)=14$
This is a model prediction, not guaranteed exact data.

The requested value is 14.

Checks and common pitfalls: This is a model prediction, not guaranteed exact data.

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

06 / 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.

07 / 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.

08 / Standard#Your turn

For Sxx=4,Syy=9,Sxy=3, find Pearson r.

S_{xx}=4;\ S_{yy}=9;\ S_{xy}=3

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

Normalize cross-variation by both coordinate scales.

Hint 2

Start from this intermediate relation.

r=S_{xy}/\sqrt{S_{xx}S_{yy}}

Worked solution

Normalize cross-variation by both coordinate scales.
Establish and simplify this mathematical relation.
$r=S_{xy}/\sqrt{S_{xx}S_{yy}}$
Establish and simplify this mathematical relation.
$r=3/6=1/2$
Regression slope 3/4 is not correlation.

The requested value is 0.5.

Checks and common pitfalls: Regression slope 3/4 is not correlation.

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

09 / Standard#Your turn

For (−1,1),(0,0),(1,1), show the correlation is zero despite a nonlinear relationship.

y=x^2;\ x=-1,0,1

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

Compute centered cross-products rather than judging the scatter as random.

Hint 2

Start from this intermediate relation.

\bar x=0;\ \bar y=2/3;\ S_{xy}=-1/3+1/3=0

Worked solution

Compute centered cross-products rather than judging the scatter as random.
Establish and simplify this mathematical relation.
$\bar x=0;\ \bar y=2/3;\ S_{xy}=-1/3+1/3=0$
Establish and simplify this mathematical relation.
$S_{xx}=2;\ S_{yy}=2/3;\ r=0$
Zero Pearson correlation excludes a linear covariance, not all dependence.

r=0; the points follow a U-shaped quadratic relation.

Checks and common pitfalls: Zero Pearson correlation excludes a linear covariance, not all dependence.

Reasoning checklist · self / teacher assessment

Compute centered cross-products rather than judging the scatter as random.
Give the intermediate relations and explain the conclusion with its restrictions.
Zero Pearson correlation excludes a linear covariance, not all dependence.

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

10 / Standard#Your turn

For pairs (0,1),(1,3),(2,2), show fitted residuals sum to zero when an intercept is fitted.

(x,y)=(0,1),(1,3),(2,2)

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

Use the fitted centroid relation.

Hint 2

Start from this intermediate relation.

\sum e_i=\sum y_i-na-b\sum x_i

Worked solution

Use the fitted centroid relation.
Establish and simplify this mathematical relation.
$\sum e_i=\sum y_i-na-b\sum x_i$
Establish and simplify this mathematical relation.
$a=\bar y-b\bar x\implies\sum e_i=0$
The residual-sum property depends on fitting an intercept.

ŷ=1.5+0.5x; residuals −0.5,1,−0.5 sum to zero.

Checks and common pitfalls: The residual-sum property depends on fitting an intercept.

Reasoning checklist · self / teacher assessment

Use the fitted centroid relation.
Give the intermediate relations and explain the conclusion with its restrictions.
The residual-sum property depends on fitting an intercept.

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

11 / Standard#Your turn

The observed x range is 1–5. Is prediction at x=100 reliable just because the line fits closely?

1\le x_{observed}\le5;\ x_{new}=100

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

Compare the prediction input with the observed range.

Hint 2

Start from this intermediate relation.

100\notin[1,5]

Worked solution

Compare the prediction input with the observed range.
Establish and simplify this mathematical relation.
$100\notin[1,5]$
Establish and simplify this mathematical relation.
$\text{in-range fit does not validate out-of-range form}$
The relation may change outside the measured range.

No; this is far extrapolation and needs separate substantive evidence.

Checks and common pitfalls: The relation may change outside the measured range.

Reasoning checklist · self / teacher assessment

Compare the prediction input with the observed range.
Give the intermediate relations and explain the conclusion with its restrictions.
The relation may change outside the measured range.

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

12 / Transfer#Your turn

Compare slopes for pairs (1,1),(2,2),(3,3) and for the same y-values reordered to (1,3),(2,2),(3,1). Explain why pairing matters.

x=(1,2,3);\ y_A=(1,2,3);\ y_B=(3,2,1)

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

Keep row identity while calculating cross-products.

Hint 2

Start from this intermediate relation.

S_{xx}=2;\ S_{xy,A}=2;\ S_{xy,B}=-2

Worked solution

Keep row identity while calculating cross-products.
Establish and simplify this mathematical relation.
$S_{xx}=2;\ S_{xy,A}=2;\ S_{xy,B}=-2$
Establish and simplify this mathematical relation.
$b_A=1;\ b_B=-1$
Sorting each column independently destroys the original association.

Slopes 1 and −1; identical marginal lists do not determine the relationship.

Checks and common pitfalls: Sorting each column independently destroys the original association.

Reasoning checklist · self / teacher assessment

Keep row identity while calculating cross-products.
Give the intermediate relations and explain the conclusion with its restrictions.
Sorting each column independently destroys the original association.

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

13 / Transfer#Your turn

For y-values 0,0,3 at x=0, compare constant predictions c=1 and c=0. Which minimizes squared error and which minimizes absolute error?

y=(0,0,3);\ \hat y=c

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

Choose the loss function before selecting a summary.

Hint 2

Start from this intermediate relation.

SSE(1)=6<SSE(0)=9

Worked solution

Choose the loss function before selecting a summary.
Establish and simplify this mathematical relation.
$SSE(1)=6<SSE(0)=9$
Establish and simplify this mathematical relation.
$SAE(0)=3<SAE(1)=4$
Least squares targets the mean; absolute-error fitting targets a median.

c=1 minimizes squared error; c=0 minimizes absolute error.

Checks and common pitfalls: Least squares targets the mean; absolute-error fitting targets a median.

Reasoning checklist · self / teacher assessment

Choose the loss function before selecting a summary.
Give the intermediate relations and explain the conclusion with its restrictions.
Least squares targets the mean; absolute-error fitting targets a median.

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

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

Choose a foundation skill to revisit ↗

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.