Kenneth ChengMathematical modeling
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LEARN · EXPLAIN · REVISE

Random sampling

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

高一必修 第二册(A版).pdf · 9.1 · PDF 180 / printed page 173

TOPIC 01

Random sampling

Build understanding of random sampling through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use random sampling 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.

Sampling design

Simple random, systematic and stratified procedures define how units are selected.

Error sources

Coverage, nonresponse and measurement problems differ from random sampling variation.

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 random sampling 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

A population has 100k students and a simple random sample has 10k. Find each student’s inclusion probability.

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
All students must have the same chance under simple random sampling.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    p=20/200=0.1p=20/200=0.1

  3. This is inclusion probability, not the fraction who will answer a survey.

The requested value is 0.1.

Checks and common pitfalls: This is inclusion probability, not the fraction who will answer a survey.

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

02 / Standard#Worked example

A stratified sample has total 10k from a school with 60k junior and 40k senior students. Find the senior sample size under proportional allocation.

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
Match the stratum’s population fraction.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    ns=20⋅80/200=8n_s=20\cdot80/200=8

  3. Stratification keeps important subgroups represented.

The requested value is 8.

Checks and common pitfalls: Stratification keeps important subgroups represented.

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

03 / Transfer#Worked example

A school studies all 100k students but measures everyone’s height with a ruler biased by +1 cm. Is there sampling error? Is there measurement bias?

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
Separate coverage from how the measurement was obtained.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    census≠error-free measurement\text{census}\ne\text{error-free measurement}

  3. A census does not remove measurement, processing or nonresponse problems.

No sampling error from selection; yes, a systematic measurement bias remains.

Checks and common pitfalls: A census does not remove measurement, processing or nonresponse problems.

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

A population has 100k students and a simple random sample has 10k. Find each student’s inclusion probability.

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
All students must have the same chance under simple random sampling.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    p=30/300=0.1p=30/300=0.1

  3. This is inclusion probability, not the fraction who will answer a survey.

The requested value is 0.1.

Checks and common pitfalls: This is inclusion probability, not the fraction who will answer a survey.

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

05 / Foundation#Your turn

A stratified sample has total 10k from a school with 60k junior and 40k senior students. Find the senior sample size under proportional allocation.

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
Match the stratum’s population fraction.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    ns=30⋅120/300=12n_s=30\cdot120/300=12

  3. Stratification keeps important subgroups represented.

The requested value is 12.

Checks and common pitfalls: Stratification keeps important subgroups represented.

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

06 / Foundation#Your turn

Systematically select 10k from an ordered population of 100k. Find the interval.

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
Divide population size by sample size.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    h=N/n=300/30=10h=N/n=300/30=10

  3. Choose a random start and inspect ordering for periodicity.

The requested value is 10.

Checks and common pitfalls: Choose a random start and inspect ordering for periodicity.

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

07 / Foundation#Your turn

A voluntary web poll receives 20k replies. Does the large reply count remove self-selection bias?

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
Separate precision from representativeness.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    large n⇏representative sample\text{large }n\not\Rightarrow\text{representative sample}

  3. A large biased sample can remain systematically misleading.

No; participation may depend on the opinion being measured.

Checks and common pitfalls: A large biased sample can remain systematically misleading.

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.

08 / Standard#Your turn

A selected sample has 10k people but only 6k reply. Find the response rate.

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
Use selected people, not the whole population, in the denominator.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    r=18/30=0.6r=18/30=0.6

  3. Nonresponse can create bias if respondents differ from nonrespondents.

The requested value is 0.6.

Checks and common pitfalls: Nonresponse can create bias if respondents differ from nonrespondents.

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

09 / Standard#Your turn

Sampling with replacement chooses 2 people from a population of k+2. Find the probability the same person appears twice.

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
Sum the mutually exclusive repeated-person outcomes.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    P=(5)(1/5)2=1/5P=(5)\left(1/5\right)^2=1/5

  3. This differs from sampling two distinct people without replacement.

The requested value is 0.2.

Checks and common pitfalls: This differs from sampling two distinct people without replacement.

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

10 / Standard#Your turn

A school studies all 100k students but measures everyone’s height with a ruler biased by +1 cm. Is there sampling error? Is there measurement bias?

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
Separate coverage from how the measurement was obtained.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    census≠error-free measurement\text{census}\ne\text{error-free measurement}

  3. A census does not remove measurement, processing or nonresponse problems.

No sampling error from selection; yes, a systematic measurement bias remains.

Checks and common pitfalls: A census does not remove measurement, processing or nonresponse problems.

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

A population has 100k students and a simple random sample has 10k. Find each student’s inclusion probability.

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
All students must have the same chance under simple random sampling.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    p=40/400=0.1p=40/400=0.1

  3. This is inclusion probability, not the fraction who will answer a survey.

The requested value is 0.1.

Checks and common pitfalls: This is inclusion probability, not the fraction who will answer a survey.

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

12 / Transfer#Your turn

A stratified sample has total 10k from a school with 60k junior and 40k senior students. Find the senior sample size under proportional allocation.

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
Match the stratum’s population fraction.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    ns=40⋅160/400=16n_s=40\cdot160/400=16

  3. Stratification keeps important subgroups represented.

The requested value is 16.

Checks and common pitfalls: Stratification keeps important subgroups represented.

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

13 / Transfer#Your turn

Systematically select 10k from an ordered population of 100k. Find the interval.

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
Identify the population and sampling mechanism before scaling counts.
Hint 2
Divide population size by sample size.
Worked solution

  1. Identify the population and sampling mechanism before scaling counts.

  2. Calculate or simplify this relation.

    h=N/n=400/40=10h=N/n=400/40=10

  3. Choose a random start and inspect ordering for periodicity.

The requested value is 10.

Checks and common pitfalls: Choose a random start and inspect ordering for periodicity.

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

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

    Board plan

    • Use random sampling to state a justified conclusion.
    • Sampling design: Simple random, systematic and stratified procedures define how units are selected.
    • Error sources: Coverage, nonresponse and measurement problems differ from random sampling variation.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Simple random, systematic and stratified procedures define how units are selected.
    • Expected reasoning: Coverage, nonresponse and measurement problems differ from random sampling variation.
    • Expected correction: A larger sample does not automatically remove selection bias.

    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.

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    Curriculum and source notes ↗