PAPER · REASONING · PRACTICE

Statistical surveys: samples, proportions and claims

Choose a defensible survey method, read proportions and state the limits of sample-based estimates.

40 minutes · Year 7, lower volume · text pp. 89–94 · textbook lessons 45–47

Textbook reference: Year 7 lower · Chapter 10 Collecting, organising and displaying data · text lessons 45–47

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

18 minutes using sampling, explicitly a synthetic population of 200 with 50% choosing A: predict random versus club-only results (3 min). Keep sample size 20 and seed 2026; compare the two selection mechanisms, recording the actual displayed counts rather than assuming them (5 min). Hold mechanism fixed and change seed to 2027, then compare a random sample of 40 with all 40 club members (6 min). Explain sampling fluctuation versus selection bias: surveying the whole biased club does not turn it into the whole school (4 min).

Organise the key idea

Choose a defensible survey method, read proportions and state the limits of sample-based estimates.

Open the original chapter Open the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

A school has 400 pupils. To study how they travel to school, randomly select 40 names from the complete pupil list and survey them. Identify the population, individual, sample and sample size.

Identify the target population and the exact way participants are selected.
Ask who has no chance or a different chance of selection.
Distinguish a sound sampling design from a guarantee that one realised sample is exact.

The target is information about every pupil, not only the 40 selected.
The sample contains the selected observations; its size counts observations.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

In a survey of 40 pupils, 12 walk, 8 cycle and 20 take a bus. Find the walking sector’s central angle.

Walking proportion: $\frac{12}{40}=\frac{3}{\square}$ .
Central angle: $\frac{12}{40}\times360^{\circ}=\square^{\circ}$ .
Check: the walking part is less than half the group, so its angle must be less than ____°.

Category	Frequency
Walk	12
Cycle	8
Bus	20

Use the stated conditions and the labelled scales.

The walking count is 12; the total is 12+8+20=40.
A complete circle represents all 40 pupils and has 360°.
Multiply 12/40 by 360; do not multiply the count 12 directly by 360.

The given categories account for every observation exactly once.
Use $\frac{12}{40}$ of the full 360° circle.
The requested result is $108$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

8 of 40 observations are category A. What percentage is A?

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The given categories account for every observation exactly once.
Use $\frac{8}{40}$ ×100% to express a percentage.
The requested result is $20$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

A frequency table has mutually exclusive counts 12,18,10 and covers everyone once. Find the total.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The given categories account for every observation exactly once.
Add the mutually exclusive category counts: $12+18+10$ .
The requested result is $40$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

To estimate the whole school’s sports preference, a learner interviews only members of the sports club. Explain a selection problem and propose a better method.

Identify the target population and the exact way participants are selected.
Ask who has no chance or a different chance of selection.
Distinguish a sound sampling design from a guarantee that one realised sample is exact.

Nonmembers have no chance of inclusion in the proposed design.
Use a complete frame and equal selection chances instead of voluntary or club-only selection.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

A survey has counts 15,9,6 for three exclusive categories. Find the first category’s relative frequency.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The given categories account for every observation exactly once.
Use $\frac{15}{30}$ as the relative frequency.
The requested result is $\frac{1}{2}$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

A random sample of 80 out of 600 pupils contains 24 walkers. Estimate the school’s walker count and label it as an estimate.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The given categories account for every observation exactly once.
Use $\frac{24}{80}$ of the stated population to obtain an estimate.
The requested result is $180$ ; this is an estimate, not a census count.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Which chart best shows how the same place’s daily temperature changes over one week?

A histogram without dates.
A line chart with ordered dates.
A sector chart treating temperatures as parts of one total.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The question asks for change over ordered time.
A line chart connects successive dates; temperatures are not parts of a total.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

To study pupil heights, 100 pupils are sampled from a school of 800. What is one individual in this investigation?

Identify the target population and the exact way participants are selected.
Ask who has no chance or a different chance of selection.
Distinguish a sound sampling design from a guarantee that one realised sample is exact.

The variable being investigated is height.
100 is sample size; one individual observation is one pupil’s height.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

A sample suggests 30% prefer A in a school of 1000, giving an estimate of 300. Must the true count equal 300? Explain.

Identify the target population and the exact way participants are selected.
Ask who has no chance or a different chance of selection.
Distinguish a sound sampling design from a guarantee that one realised sample is exact.

The estimate uses the sample proportion.
A possible true count such as 320 is not ruled out by that estimate alone.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Counts A=6,B=9,C=15 cover the whole group. Find A’s central angle.

Use the stated conditions and the labelled scales.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The given categories account for every observation exactly once.
Use $\frac{6}{30}$ of the full 360° circle.
The requested result is $72$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

A table totals 50 observations; known counts are 12 and 18, with one missing category. Find the missing count.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Subtract known counts from the total.
50−12−18=20.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

Testing the lifetime of a bulb destroys it. Should a factory test every saleable bulb to estimate lifetime? Give a suitable method.

Identify the target population and the exact way participants are selected.
Ask who has no chance or a different chance of selection.
Distinguish a sound sampling design from a guarantee that one realised sample is exact.

Consider the purpose and cost of a census.
Random selection across the production batch is preferable to selecting only one convenient box.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Two percentages are 40% and 50%. A chart begins its vertical axis at 39%, making visible bar heights 1 and 11. Does this mean the second percentage is eleven times the first?

Identify the target population and the exact way participants are selected.
Ask who has no chance or a different chance of selection.
Distinguish a sound sampling design from a guarantee that one realised sample is exact.

The visible heights are distances above 39, not the complete percentages.
Use the original values when forming a ratio.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

Three age groups have population ratio 3:4:3. Choose a sample of 100 proportionally, then randomly within each group. How many from each? Does proportional allocation alone eliminate every possible bias?

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The ratio totals 10 parts, so each part is 10 people.
Allocate 30,40,30; use a clear random mechanism within each complete group list.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

A synthetic population has 1000 members, 500 choosing A. Two random samples of 40 contain 19 and 21 A choices. Must one sample have been selected incorrectly? Explain.

Identify the target population and the exact way participants are selected.
Ask who has no chance or a different chance of selection.
Distinguish a sound sampling design from a guarantee that one realised sample is exact.

Each count is possible given 500 A and 500 non-A members.
Repeat samples can vary; inspect the selection mechanism separately.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Preparation and assessment notes

Prerequisites

Read a frequency table and calculate fractions, percentages and a part of 360 degrees.

Look for in students’ work

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original 40-minute core consolidation after the mapped textbook lessons; not a replacement for every first-teaching period. Only the supplied text has been mapped.

大湾区初中数学三维课堂（七年级下册）正文.pdf

Year 7, lower volume · text pp. 89–94 · textbook lessons 45–47

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Statistical surveys: samples, proportions and claims

Year 7, lower volume · text pp. 89–94 · textbook lessons 45–47 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

A school has 400 pupils. To study how they travel to school, randomly select 40 names from the complete pupil list and survey them. Identify the population, individual, sample and sample size.

Guided attempt2

In a survey of 40 pupils, 12 walk, 8 cycle and 20 take a bus. Find the walking sector’s central angle.

Walking proportion: $\frac{12}{40}=\frac{3}{\square}$ .
Central angle: $\frac{12}{40}\times360^{\circ}=\square^{\circ}$ .
Check: the walking part is less than half the group, so its angle must be less than ____°.

Category	Frequency
Walk	12
Cycle	8
Bus	20

Use the stated conditions and the labelled scales.

Prerequisite3

8 of 40 observations are category A. What percentage is A?

Prerequisite4

A frequency table has mutually exclusive counts 12,18,10 and covers everyone once. Find the total.

Statistical surveys: samples, proportions and claims

Year 7, lower volume · text pp. 89–94 · textbook lessons 45–47 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

To estimate the whole school’s sports preference, a learner interviews only members of the sports club. Explain a selection problem and propose a better method.

Independent practice6

A survey has counts 15,9,6 for three exclusive categories. Find the first category’s relative frequency.

Independent practice7

A random sample of 80 out of 600 pupils contains 24 walkers. Estimate the school’s walker count and label it as an estimate.

Independent practice8

Which chart best shows how the same place’s daily temperature changes over one week?

A histogram without dates.
A line chart with ordered dates.
A sector chart treating temperatures as parts of one total.

Exit question9

To study pupil heights, 100 pupils are sampled from a school of 800. What is one individual in this investigation?

Exit question10

A sample suggests 30% prefer A in a school of 1000, giving an estimate of 300. Must the true count equal 300? Explain.

Statistical surveys: samples, proportions and claims — Further practice

Year 7, lower volume · text pp. 89–94 · textbook lessons 45–47

Further practice11

Counts A=6,B=9,C=15 cover the whole group. Find A’s central angle.

Use the stated conditions and the labelled scales.

Further practice12

A table totals 50 observations; known counts are 12 and 18, with one missing category. Find the missing count.

Further practice13

Testing the lifetime of a bulb destroys it. Should a factory test every saleable bulb to estimate lifetime? Give a suitable method.

Further practice14

Two percentages are 40% and 50%. A chart begins its vertical axis at 39%, making visible bar heights 1 and 11. Does this mean the second percentage is eleven times the first?

Extension15

Extension16

A synthetic population has 1000 members, 500 choosing A. Two random samples of 40 contain 19 and 21 A choices. Must one sample have been selected incorrectly? Explain.

Statistical surveys: samples, proportions and claims — Teacher answers

Year 7, lower volume · text pp. 89–94 · textbook lessons 45–47

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Read a frequency table and calculate fractions, percentages and a part of 360 degrees.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

Inquiry and discussion: 18 minutes using sampling, explicitly a synthetic population of 200 with 50% choosing A: predict random versus club-only results (3 min). Keep sample size 20 and seed 2026; compare the two selection mechanisms, recording the actual displayed counts rather than assuming them (5 min). Hold mechanism fixed and change seed to 2027, then compare a random sample of 40 with all 40 club members (6 min). Explain sampling fluctuation versus selection bias: surveying the whole biased club does not turn it into the whole school (4 min).

Worked example1

A school has 400 pupils. To study how they travel to school, randomly select 40 names from the complete pupil list and survey them. Identify the population, individual, sample and sample size.

Reference answer

Population: all 400 pupils’ travel information; individual: one pupil’s travel information; sample: the 40 selected pupils’ travel information; sample size: 40.

The target is information about every pupil, not only the 40 selected.
The sample contains the selected observations; its size counts observations.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

In a survey of 40 pupils, 12 walk, 8 cycle and 20 take a bus. Find the walking sector’s central angle.

Walking proportion: $\frac{12}{40}=\frac{3}{\square}$ .
Central angle: $\frac{12}{40}\times360^{\circ}=\square^{\circ}$ .
Check: the walking part is less than half the group, so its angle must be less than ____°.

Category	Frequency
Walk	12
Cycle	8
Bus	20

Use the stated conditions and the labelled scales.

Reference answer

$108$ °.

(12/40)\cdot 360

The given categories account for every observation exactly once.
Use $\frac{12}{40}$ of the full 360° circle.
The requested result is $108$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Category	Frequency
Walk	12
Cycle	8
Bus	20

Use the stated conditions and the labelled scales.

Prerequisite3

8 of 40 observations are category A. What percentage is A?

Reference answer

$20$ %.

(8/40)\cdot 100

The given categories account for every observation exactly once.
Use $\frac{8}{40}$ ×100% to express a percentage.
The requested result is $20$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

A frequency table has mutually exclusive counts 12,18,10 and covers everyone once. Find the total.

Reference answer

$40$ .

12+18+10

The given categories account for every observation exactly once.
Add the mutually exclusive category counts: $12+18+10$ .
The requested result is $40$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

To estimate the whole school’s sports preference, a learner interviews only members of the sports club. Explain a selection problem and propose a better method.

Reference answer

The club may overrepresent sports enthusiasts. Randomly select from the complete school list; a random sample can still fluctuate.

Nonmembers have no chance of inclusion in the proposed design.
Use a complete frame and equal selection chances instead of voluntary or club-only selection.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

A survey has counts 15,9,6 for three exclusive categories. Find the first category’s relative frequency.

Reference answer

$\frac{1}{2}$ .

(15/30)\cdot 1

The given categories account for every observation exactly once.
Use $\frac{15}{30}$ as the relative frequency.
The requested result is $\frac{1}{2}$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

A random sample of 80 out of 600 pupils contains 24 walkers. Estimate the school’s walker count and label it as an estimate.

Reference answer

Estimated $180$ pupils.

(24/80)\cdot 600

The given categories account for every observation exactly once.
Use $\frac{24}{80}$ of the stated population to obtain an estimate.
The requested result is $180$ ; this is an estimate, not a census count.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Which chart best shows how the same place’s daily temperature changes over one week?

A histogram without dates.
A line chart with ordered dates.
A sector chart treating temperatures as parts of one total.

Reference answer

A line chart with ordered dates.

The question asks for change over ordered time.
A line chart connects successive dates; temperatures are not parts of a total.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

To study pupil heights, 100 pupils are sampled from a school of 800. What is one individual in this investigation?

Reference answer

One pupil’s height observation, not the number 100.

The variable being investigated is height.
100 is sample size; one individual observation is one pupil’s height.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

A sample suggests 30% prefer A in a school of 1000, giving an estimate of 300. Must the true count equal 300? Explain.

Reference answer

No. A sample estimate can differ from the population count.

The estimate uses the sample proportion.
A possible true count such as 320 is not ruled out by that estimate alone.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Counts A=6,B=9,C=15 cover the whole group. Find A’s central angle.

Use the stated conditions and the labelled scales.

Reference answer

$72$ °.

(6/30)\cdot 360

The given categories account for every observation exactly once.
Use $\frac{6}{30}$ of the full 360° circle.
The requested result is $72$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the stated conditions and the labelled scales.

Further practice12

A table totals 50 observations; known counts are 12 and 18, with one missing category. Find the missing count.

Reference answer

$20$ .

50-12-18

Subtract known counts from the total.
50−12−18=20.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

Testing the lifetime of a bulb destroys it. Should a factory test every saleable bulb to estimate lifetime? Give a suitable method.

Reference answer

Use a properly selected sample; a destructive census would leave no tested bulbs saleable.

Consider the purpose and cost of a census.
Random selection across the production batch is preferable to selecting only one convenient box.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Two percentages are 40% and 50%. A chart begins its vertical axis at 39%, making visible bar heights 1 and 11. Does this mean the second percentage is eleven times the first?

Reference answer

No. The real ratio is 50/40=5/4, and the truncated baseline exaggerates the visual difference.

The visible heights are distances above 39, not the complete percentages.
Use the original values when forming a ratio.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

Reference answer

30,40,30; no. Selection and nonresponse within each group still matter.

The ratio totals 10 parts, so each part is 10 people.
Allocate 30,40,30; use a clear random mechanism within each complete group list.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

A synthetic population has 1000 members, 500 choosing A. Two random samples of 40 contain 19 and 21 A choices. Must one sample have been selected incorrectly? Explain.

Reference answer

No. Both are feasible random outcomes around the population proportion 1/2; variation alone is not proof of biased selection.

Each count is possible given 500 A and 500 non-A members.
Repeat samples can vary; inspect the selection mechanism separately.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.