Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Statistical case study

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

高一必修 第二册(A版).pdf · 9.3 · PDF 227 / printed page 220

Revisit first: Estimating a population from a sample

TOPIC 01

Statistical case study

Build understanding of statistical case study through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use statistical case study to state a justified conclusion.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Full investigation

Define the target, collect compatible data, analyze and report limitations.

Ethical reporting

Use anonymous classroom data and preserve uncertainty and exclusions.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in statistical case study changes. Record a reason.

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: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

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

An anonymous commute survey records 20k students; 12k travel at most 30 minutes. Find the sample proportion.

k=2k=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Count the students meeting the stated inclusive threshold.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    p^=24/40=0.6\hat p=24/40=0.6

  3. The measured outcome is duration, not distance.

The requested value is 0.6.

Checks and common pitfalls: The measured outcome is duration, not distance.

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

02 / Standard#Worked example

Two strata have mean commute times 20 and 40 minutes and sizes 3k and k. Find the overall mean.

k=2k=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Weight stratum means by their sizes.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    xˉ=[20(6)+40(2)]/8=25\bar x=[20(6)+40(2)]/8=25

  3. A simple average of the two means would incorrectly ignore the unequal sizes.

The requested value is 25.

Checks and common pitfalls: A simple average of the two means would incorrectly ignore the unequal sizes.

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

03 / Transfer#Worked example

A report gives only the estimated proportion k/(k+2). Name two additional pieces of information needed to judge it.

k=2k=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Ask how the observations were selected and how many contributed.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    estimate+design+sample size\text{estimate} + \text{design} + \text{sample size}

  3. A percentage alone does not reveal representativeness or precision.

Sampling method and sample size, with the target population and response rate also relevant.

Checks and common pitfalls: A percentage alone does not reveal representativeness or precision.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

04 / Foundation#Your turn

An anonymous commute survey records 20k students; 12k travel at most 30 minutes. Find the sample proportion.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Count the students meeting the stated inclusive threshold.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    p^=36/60=0.6\hat p=36/60=0.6

  3. The measured outcome is duration, not distance.

The requested value is 0.6.

Checks and common pitfalls: The measured outcome is duration, not distance.

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

05 / Foundation#Your turn

Two strata have mean commute times 20 and 40 minutes and sizes 3k and k. Find the overall mean.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Weight stratum means by their sizes.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    xˉ=[20(9)+40(3)]/12=25\bar x=[20(9)+40(3)]/12=25

  3. A simple average of the two means would incorrectly ignore the unequal sizes.

The requested value is 25.

Checks and common pitfalls: A simple average of the two means would incorrectly ignore the unequal sizes.

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

06 / Foundation#Your turn

A questionnaire asks only classmates arriving before 8 am. Explain a coverage issue when estimating the whole school’s commute time.

n=30n=30
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Compare the sampling frame with the target population.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    sampling frame⊊target population\text{sampling frame}\subsetneq\text{target population}

  3. More responses within the same restricted frame do not repair that exclusion.

Late-arriving students are excluded and may have systematically different commutes.

Checks and common pitfalls: More responses within the same restricted frame do not repair that exclusion.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

07 / Foundation#Your turn

The class counts for commute time <20,20–40,>40 minutes are k,2k,k. Find the fraction above 40.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Use the relevant class and the total count.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    p=3/12=1/4p=3/12=1/4

  3. Do not merge the middle class unless the threshold requires it.

The requested value is 0.25.

Checks and common pitfalls: Do not merge the middle class unless the threshold requires it.

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

08 / Standard#Your turn

Survey responses 5k contain 2k reports in seconds although the question asked minutes. Is direct pooling valid?

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Check definitions and units before summary statistics.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    1 minute=60 seconds1\text{ minute}=60\text{ seconds}

  3. Cleaning decisions should remain documented, not silently delete difficult cases.

No. Convert compatible units and flag ambiguous responses before analysis.

Checks and common pitfalls: Cleaning decisions should remain documented, not silently delete difficult cases.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

09 / Standard#Your turn

A survey finds longer commute is associated with lower attendance. Does the survey prove that commute caused the attendance pattern?

n=60n=60
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Distinguish the observation from a causal claim.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    association⇏causation\text{association}\not\Rightarrow\text{causation}

  3. Use a conclusion proportional to the survey design.

No. School location, transport reliability or other factors may confound the relation.

Checks and common pitfalls: Use a conclusion proportional to the survey design.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

10 / Standard#Your turn

A report gives only the estimated proportion k/(k+2). Name two additional pieces of information needed to judge it.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Ask how the observations were selected and how many contributed.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    estimate+design+sample size\text{estimate} + \text{design} + \text{sample size}

  3. A percentage alone does not reveal representativeness or precision.

Sampling method and sample size, with the target population and response rate also relevant.

Checks and common pitfalls: A percentage alone does not reveal representativeness or precision.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

11 / Standard#Your turn

An anonymous commute survey records 20k students; 12k travel at most 30 minutes. Find the sample proportion.

k=4k=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Count the students meeting the stated inclusive threshold.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    p^=48/80=0.6\hat p=48/80=0.6

  3. The measured outcome is duration, not distance.

The requested value is 0.6.

Checks and common pitfalls: The measured outcome is duration, not distance.

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

12 / Transfer#Your turn

Two strata have mean commute times 20 and 40 minutes and sizes 3k and k. Find the overall mean.

k=4k=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Weight stratum means by their sizes.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    xˉ=[20(12)+40(4)]/16=25\bar x=[20(12)+40(4)]/16=25

  3. A simple average of the two means would incorrectly ignore the unequal sizes.

The requested value is 25.

Checks and common pitfalls: A simple average of the two means would incorrectly ignore the unequal sizes.

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

13 / Transfer#Your turn

A questionnaire asks only classmates arriving before 8 am. Explain a coverage issue when estimating the whole school’s commute time.

n=40n=40
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
State the survey question, population, sampling and measurement rules before drawing a conclusion.
Hint 2
Compare the sampling frame with the target population.
Worked solution

  1. State the survey question, population, sampling and measurement rules before drawing a conclusion.

  2. Calculate or simplify this relation.

    sampling frame⊊target population\text{sampling frame}\subsetneq\text{target population}

  3. More responses within the same restricted frame do not repair that exclusion.

Late-arriving students are excluded and may have systematically different commutes.

Checks and common pitfalls: More responses within the same restricted frame do not repair that exclusion.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

Focus on one question

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

    Question sequence

    • What must be true before using the main rule for statistical case study?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use statistical case study to state a justified conclusion.
    • Full investigation: Define the target, collect compatible data, analyze and report limitations.
    • Ethical reporting: Use anonymous classroom data and preserve uncertainty and exclusions.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Define the target, collect compatible data, analyze and report limitations.
    • Expected reasoning: Use anonymous classroom data and preserve uncertainty and exclusions.
    • Expected correction: A statistical association alone is not a causal conclusion.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

    No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

    Curriculum and source notes ↗