Kenneth ChengMathematical modeling
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Estimating a population from a sample

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

高一必修 第二册(A版).pdf · 9.2 · PDF 200 / printed page 193

Revisit first: Random sampling

TOPIC 01

Estimating a population from a sample

Build understanding of estimating a population from a sample through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use estimating a population from a sample 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.

Distribution estimates

Use frequency, class density and sample proportions to describe data.

Location and spread

Mean, median, quantiles and variance answer different questions about the distribution.

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 estimating a population from a sample 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

In a sample of 20k, 5k use public transport. Estimate the population 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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Use the observed fraction as an estimate.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    p^=10/40=0.25\hat p=10/40=0.25

  3. The population proportion is estimated rather than known exactly.

The requested value is 0.25.

Checks and common pitfalls: The population proportion is estimated rather than known exactly.

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

02 / Standard#Worked example

A representative sample estimates a proportion 1/4 in a population of 80k. Estimate the corresponding count.

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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Multiply the estimated fraction by the population size.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    N^=160/4=40\hat N=160/4=40

  3. The estimated count inherits sampling and coverage uncertainty.

The requested value is 40.

Checks and common pitfalls: The estimated count inherits sampling and coverage uncertainty.

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

03 / Transfer#Worked example

A sample’s mean is k and one extreme observation 100k is added. Must the median change as much as the mean? Explain.

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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Compare value weighting with order statistics.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    xˉnew=(nxˉ+100k)/(n+1)\bar x_{new}=(n\bar x+100k)/(n+1)

  3. The change in the median depends on the ordered data; it cannot be inferred from the mean alone.

No. The mean uses numerical magnitudes; the median mainly depends on ranks.

Checks and common pitfalls: The change in the median depends on the ordered data; it cannot be inferred from the mean alone.

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

In a sample of 20k, 5k use public transport. Estimate the population 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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Use the observed fraction as an estimate.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    p^=15/60=0.25\hat p=15/60=0.25

  3. The population proportion is estimated rather than known exactly.

The requested value is 0.25.

Checks and common pitfalls: The population proportion is estimated rather than known exactly.

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

05 / Foundation#Your turn

A representative sample estimates a proportion 1/4 in a population of 80k. Estimate the corresponding count.

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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Multiply the estimated fraction by the population size.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    N^=240/4=60\hat N=240/4=60

  3. The estimated count inherits sampling and coverage uncertainty.

The requested value is 60.

Checks and common pitfalls: The estimated count inherits sampling and coverage uncertainty.

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

06 / Foundation#Your turn

A histogram class [0,2k) has frequency 6k. Find its frequency density (count per unit).

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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Divide frequency by class width.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    d=18/(6−0)=3d=18/(6-0)=3

  3. With unequal widths, area represents frequency; height alone does not.

The requested value is 3.

Checks and common pitfalls: With unequal widths, area represents frequency; height alone does not.

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

07 / Foundation#Your turn

For observations k,k+2,k+4, find the sample 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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Add all observations and divide by their count.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    xˉ=(3+5+7)/3=5\bar x=(3+5+7)/3=5

  3. Every observation is equally weighted in this unweighted sample.

The requested value is 5.

Checks and common pitfalls: Every observation is equally weighted in this unweighted sample.

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

08 / Standard#Your turn

For observations k,k+2,k+4, find variance using denominator 3 as specified.

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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Center the data before squaring; use the stated denominator.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    v=[(−2)2+02+22]/3=8/3v=[(-2)^2+0^2+2^2]/3=8/3

  3. This descriptive variance differs from an unbiased sample variance using n−1.

The requested value is 2.66666667.

Checks and common pitfalls: This descriptive variance differs from an unbiased sample variance using n−1.

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

09 / Standard#Your turn

The sample proportions for three exhaustive categories are 1/4,1/2,p. Find p.

n=12n=12
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
The categories partition the sample.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    p=1−1/4−1/2=1/4p=1-1/4-1/2=1/4

  3. Check that the categories are disjoint before adding their proportions.

The requested value is 0.25.

Checks and common pitfalls: Check that the categories are disjoint before adding their proportions.

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

10 / Standard#Your turn

A sample’s mean is k and one extreme observation 100k is added. Must the median change as much as the mean? Explain.

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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Compare value weighting with order statistics.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    xˉnew=(nxˉ+100k)/(n+1)\bar x_{new}=(n\bar x+100k)/(n+1)

  3. The change in the median depends on the ordered data; it cannot be inferred from the mean alone.

No. The mean uses numerical magnitudes; the median mainly depends on ranks.

Checks and common pitfalls: The change in the median depends on the ordered data; it cannot be inferred from the mean alone.

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

In a sample of 20k, 5k use public transport. Estimate the population 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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Use the observed fraction as an estimate.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    p^=20/80=0.25\hat p=20/80=0.25

  3. The population proportion is estimated rather than known exactly.

The requested value is 0.25.

Checks and common pitfalls: The population proportion is estimated rather than known exactly.

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

12 / Transfer#Your turn

A representative sample estimates a proportion 1/4 in a population of 80k. Estimate the corresponding count.

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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Multiply the estimated fraction by the population size.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    N^=320/4=80\hat N=320/4=80

  3. The estimated count inherits sampling and coverage uncertainty.

The requested value is 80.

Checks and common pitfalls: The estimated count inherits sampling and coverage uncertainty.

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

13 / Transfer#Your turn

A histogram class [0,2k) has frequency 6k. Find its frequency density (count per unit).

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
Translate counts into sample proportions and keep classes, units and uncertainty visible.
Hint 2
Divide frequency by class width.
Worked solution

  1. Translate counts into sample proportions and keep classes, units and uncertainty visible.

  2. Calculate or simplify this relation.

    d=24/(8−0)=3d=24/(8-0)=3

  3. With unequal widths, area represents frequency; height alone does not.

The requested value is 3.

Checks and common pitfalls: With unequal widths, area represents frequency; height alone does not.

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 estimating a population from a sample?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use estimating a population from a sample to state a justified conclusion.
    • Distribution estimates: Use frequency, class density and sample proportions to describe data.
    • Location and spread: Mean, median, quantiles and variance answer different questions about the distribution.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Use frequency, class density and sample proportions to describe data.
    • Expected reasoning: Mean, median, quantiles and variance answer different questions about the distribution.
    • Expected correction: A sample estimate is not a known exact population parameter.

    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 ↗