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Statistical analysis of paired data: mixed review

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

TOPIC 01

Statistical analysis of paired data: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

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

Find r from centered summaries.

S_{xx}=4,\quad S_{yy}=400,\quad S_{xy}=20

Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.

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.

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

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$r=20/(40)=1/2$
The positive square root gives the denominator scale.

The requested value is 0.5.

Checks and common pitfalls: The positive square root gives the denominator scale.

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

02 / Foundation#Your turn

Find the intercept.

\bar x=2,\quad\bar y=45,\quad b=21

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=\bar y-b\bar x

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$a=45-2(21)=3$
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.

03 / Foundation#Your turn

Calculate χ² for the table.

O=\begin{pmatrix}64&32\\32&64\end{pmatrix}

Observations must be independent; expected counts support the supplied large-sample approximation. Rejection is evidence of association, not causation.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

All expected cells are 3u/2.

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$\chi^2=4[(32/2)^2/(96/2)]=64/3$
All four nonnegative cell contributions must be included.

The requested value is 21.3333333333.

Checks and common pitfalls: All four nonnegative cell contributions must be included.

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

04 / Foundation#Your turn

For pairs (−1,t),(0,0),(1,t), find Sxy.

t=23

Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.

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.

\bar x=0;\quad S_{xy}=\sum x_i(y_i-\bar y)

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$S_{xy}=-1(23-46/3)+1(23-46/3)=0$
Symmetry cancels the centered products even though the pattern is nonlinear.

The requested value is 0.

Checks and common pitfalls: Symmetry cancels the centered products even though the pattern is nonlinear.

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

05 / Foundation#Your turn

Find the fitted slope.

S_{xx}=2,\quad S_{xy}=48

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=S_{xy}/S_{xx}

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$b=48/2=24$
The denominator is the variation in the predictor x.

The requested value is 24.

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

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

06 / Foundation#Your turn

Under independence, find E11.

O=\begin{pmatrix}70&35\\35&70\end{pmatrix}

Observations must be independent; expected counts support the supplied large-sample approximation. Rejection is evidence of association, not causation.

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_{11}=R_1C_1/N

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$E_{11}=(105)^2/210=52.5$
Use both relevant margins and the full sample size.

The requested value is 52.5.

Checks and common pitfalls: Use both relevant margins and the full sample size.

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

07 / Standard#Your turn

For pairs (−1,t),(0,0),(1,t), find Sxy.

t=26

Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.

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.

\bar x=0;\quad S_{xy}=\sum x_i(y_i-\bar y)

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$S_{xy}=-1(26-52/3)+1(26-52/3)=0$
Symmetry cancels the centered products even though the pattern is nonlinear.

The requested value is 0.

Checks and common pitfalls: Symmetry cancels the centered products even though the pattern is nonlinear.

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

08 / Standard#Your turn

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

t=27

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.

\bar x=\bar y=0

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$S_{xx}=2,\quad S_{xy}=54$
Apply the stated relation and retain its conditions.
$b=27$
Preserve the paired order when calculating products.

The requested value is 27.

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

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

09 / Standard#Your turn

A before/after category table records the same t people twice. Can a standard independent-observation chi-square analysis treat the records as separate people?

t=38

Observations must be independent; expected counts support the supplied large-sample approximation. Rejection is evidence of association, not causation.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Check whether rows share participants.

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$\text{paired records are dependent}$
The study needs a method for paired categorical outcomes rather than treating repeated people as independent.

The requested relation or conclusion is shown below.

\text{the independent-observation assumption fails}

Checks and common pitfalls: The study needs a method for paired categorical outcomes rather than treating repeated people as independent.

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.

10 / Standard#Your turn

For pairs (−1,t),(0,0),(1,t), find Sxy.

t=29

Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.

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.

\bar x=0;\quad S_{xy}=\sum x_i(y_i-\bar y)

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$S_{xy}=-1(29-58/3)+1(29-58/3)=0$
Symmetry cancels the centered products even though the pattern is nonlinear.

The requested value is 0.

Checks and common pitfalls: Symmetry cancels the centered products even though the pattern is nonlinear.

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 least-squares slope.

t=30

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.

\bar x=\bar y=0

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$S_{xx}=2,\quad S_{xy}=60$
Apply the stated relation and retain its conditions.
$b=30$
Preserve the paired order when calculating products.

The requested value is 30.

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

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

12 / Standard#Your turn

A before/after category table records the same t people twice. Can a standard independent-observation chi-square analysis treat the records as separate people?

t=41

Observations must be independent; expected counts support the supplied large-sample approximation. Rejection is evidence of association, not causation.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Check whether rows share participants.

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$\text{paired records are dependent}$
The study needs a method for paired categorical outcomes rather than treating repeated people as independent.

The requested relation or conclusion is shown below.

\text{the independent-observation assumption fails}

Checks and common pitfalls: The study needs a method for paired categorical outcomes rather than treating repeated people as independent.

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

An observational survey reports the displayed positive correlation between study time and score. Does it prove a causal effect?

r=0.1

Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Consider confounding and study design.

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$\text{Prior preparation may influence both variables.}$
The size of an observational correlation does not replace a causal design or justified control of confounding.

The requested relation or conclusion is shown below.

\text{association does not establish causation}

Checks and common pitfalls: The size of an observational correlation does not replace a causal design or justified control of confounding.

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.

14 / Transfer#Your turn

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

t=33

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.

S_{xy}=0,\quad \bar y=2t/3

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$b=0,\quad a=66/3$
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.

\hat y=66/3

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.

15 / Transfer#Your turn

A before/after category table records the same t people twice. Can a standard independent-observation chi-square analysis treat the records as separate people?

t=44

Observations must be independent; expected counts support the supplied large-sample approximation. Rejection is evidence of association, not causation.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Check whether rows share participants.

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$\text{paired records are dependent}$
The study needs a method for paired categorical outcomes rather than treating repeated people as independent.

The requested relation or conclusion is shown below.

\text{the independent-observation assumption fails}

Checks and common pitfalls: The study needs a method for paired categorical outcomes rather than treating repeated people as independent.

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.

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.

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Choose a foundation skill to revisit ↗

Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.