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Statistics: mixed assessment

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

TOPIC 01

Statistics: mixed assessment

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

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

k=7

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

Identify the population and sampling mechanism before scaling counts.
Calculate or simplify this relation.
$p=70/700=0.1$
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 / Foundation#Your turn

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

k=7

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

Translate counts into sample proportions and keep classes, units and uncertainty visible.
Calculate or simplify this relation.
$\hat N=560/4=140$
The estimated count inherits sampling and coverage uncertainty.

The requested value is 140.

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

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

03 / 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=70

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

State the survey question, population, sampling and measurement rules before drawing a conclusion.
Calculate or simplify this relation.
$\text{sampling frame}\subsetneq\text{target population}$
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.

04 / Foundation#Your turn

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

k=8

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

Identify the population and sampling mechanism before scaling counts.
Calculate or simplify this relation.
$\text{large }n\not\Rightarrow\text{representative sample}$
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.

05 / Foundation#Your turn

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

k=8

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

Translate counts into sample proportions and keep classes, units and uncertainty visible.
Calculate or simplify this relation.
$v=[(-2)^2+0^2+2^2]/3=8/3$
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.

06 / Foundation#Your turn

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

n=160

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

State the survey question, population, sampling and measurement rules before drawing a conclusion.
Calculate or simplify this relation.
$\text{association}\not\Rightarrow\text{causation}$
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.

07 / 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=9

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

Identify the population and sampling mechanism before scaling counts.
Calculate or simplify this relation.
$\text{census}\ne\text{error-free measurement}$
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.

08 / Standard#Your turn

In a sample of 20k, 5k use public transport. Estimate the population proportion.

k=9

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

Translate counts into sample proportions and keep classes, units and uncertainty visible.
Calculate or simplify this relation.
$\hat p=45/180=0.25$
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.

09 / Standard#Your turn

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

k=9

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

State the survey question, population, sampling and measurement rules before drawing a conclusion.
Calculate or simplify this relation.
$\bar x=[20(27)+40(9)]/36=25$
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.

10 / Standard#Your turn

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

k=10

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

Identify the population and sampling mechanism before scaling counts.
Calculate or simplify this relation.
$h=N/n=1000/100=10$
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.

11 / Standard#Your turn

For observations k,k+2,k+4, find the sample mean.

k=10

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

Translate counts into sample proportions and keep classes, units and uncertainty visible.
Calculate or simplify this relation.
$\bar x=(10+12+14)/3=12$
Every observation is equally weighted in this unweighted sample.

The requested value is 12.

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

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

12 / Standard#Your turn

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

k=10

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

State the survey question, population, sampling and measurement rules before drawing a conclusion.
Calculate or simplify this relation.
$1\text{ minute}=60\text{ seconds}$
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.

13 / Transfer#Your turn

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

k=11

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

Identify the population and sampling mechanism before scaling counts.
Calculate or simplify this relation.
$P=(13)\left(1/13\right)^2=1/13$
This differs from sampling two distinct people without replacement.

The requested value is 0.07692308.

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

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

14 / Transfer#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=11

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

Translate counts into sample proportions and keep classes, units and uncertainty visible.
Calculate or simplify this relation.
$\bar x_{new}=(n\bar x+100k)/(n+1)$
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.

15 / Transfer#Your turn

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

k=11

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

State the survey question, population, sampling and measurement rules before drawing a conclusion.
Calculate or simplify this relation.
$\hat p=132/220=0.6$
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.

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.