Year 7 Mathematics · AC9M7ST03

Statistical Investigations and Sampling

plan and conduct statistical investigations involving data for discrete and continuous numerical variables; analyse and interpret distributions of data and report…

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Learning goalsSay it simply

We are learning to plan and conduct statistical investigations and communicate findings that are justified by the sampled data.

A statistical investigation begins with a question that anticipates variation, identifies the population of interest and defines a measurable numerical variable. The sample, measurement procedure and units must match that question so the collected data can actually address it.

A representative sample depends on selection method rather than size alone. Simple random or carefully implemented systematic sampling can reduce selection bias, while convenience and voluntary-response samples often over-represent accessible or strongly motivated people; nonresponse can introduce further bias.

After collection, data are checked, displayed and summarised by centre, spread and shape before a conclusion is written in context. The conclusion should distinguish the observed sample from the broader population, disclose limitations and avoid causal language unless the investigation design supports cause.

Success criteria

  • I can write a statistical question and define its population, sample and numerical variable.
  • I can select a feasible sampling and data-collection method while identifying possible bias and nonresponse.
  • I can analyse a distribution and write a contextual conclusion with appropriate limitations.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Planning an investigation of Year 7 travel times.

  1. Ask, 'How long does it usually take students in our Year 7 cohort to travel to school?' and define the population as all Year 7 students at the school.
  2. Select a random sample from every Year 7 home group so that one accessible class does not dominate, and record travel time in minutes using the same wording and reference day.
  3. Check missing, impossible and duplicate entries, then display the continuous numerical data and calculate an appropriate centre and spread.
  4. If the sample has median 18 minutes, range 52 minutes and a longer upper tail, conclude that a typical sampled journey was about 18 minutes but travel times varied widely, without claiming an exact result for every student.

Final answer: The investigation links a variable statistical question to a distributed random sample, consistent collection, distribution analysis and a qualified conclusion about Year 7 travel times.

Check: Every conclusion refers to the sampled travel-time data, reports variation and avoids extending the result beyond the defined school population.

Example 2

Evaluating a biased survey of physical activity.

  1. A student surveys 120 members of school sports clubs to estimate weekly activity for all 600 Year 7 students.
  2. Although the sample is large, club membership makes highly active students more likely to be included, so the sample is a convenience sample with selection bias.
  3. Improve the design by randomly selecting students from the full Year 7 roll, or by randomly sampling the same proportion from each home group, and follow up nonresponses consistently.
  4. Report the sample size, response rate, activity distribution and remaining limitations before generalising cautiously to the Year 7 population.

Final answer: The sports-club survey cannot represent all Year 7 activity reliably, and a random sample from the full cohort with transparent nonresponse reporting is stronger.

Check: The revised sampling frame gives every Year 7 student a known opportunity for selection rather than restricting selection to sports-club members.

Example 3

Application problem 1

Problem: A school surveys only students in the library at lunchtime about preferred sports. Identify the sampling problem and propose a better method for estimating the whole school's preference.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: The sample is biased because library users may not represent the whole school. A random or stratified sample across year levels and locations would better represent the population.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: The sample is biased because library users may not represent the whole school. A random or stratified sample across year levels and locations would better represent the population.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: Design a feasible investigation of Year 7 sleep duration that states a statistical question, population, sample, variable, collection method and one limitation.

  1. Plan: Write one clause for each required component and make the reference night consistent for every sampled student.
  2. Work: Ask 'How many hours do students in our Year 7 cohort usually sleep on a school night?', define all Year 7 students as the population, randomly sample students from every home group, collect self-reported hours for the same specified night, analyse the distribution, and acknowledge that recall and self-reporting may be inaccurate.
  3. Interpret: The plan aligns every investigation component and identifies a realistic measurement limitation.

Final answer: Ask 'How many hours do students in our Year 7 cohort usually sleep on a school night?', define all Year 7 students as the population, randomly sample students from every home group, collect self-reported hours for the same specified night, analyse the distribution, and acknowledge that recall and self-reporting may be inaccurate.

Check: The plan aligns every investigation component and identifies a realistic measurement limitation.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7ST03 - Statistical Investigations and Sampling
Example 1

Example 1 Planning an investigation of Year 7 travel times. Ask, 'How long does it usually take students in our Year 7 cohort to travel to school?' and define the population as all Year 7 students at the school. Select a random sample from every Year 7 home group so that one accessible class does not dominate, and record travel time in minutes using the same wording and reference day. Check missing, impossible and duplicate entries, then display the continuous numerical data and calculate an appropriate centre and spread. If the sample has median 18 minutes, range 52 minutes and a longer upper tail, conclude that a typical sampled journey was about 18 minutes but travel times varied widely, without claiming an exact result for every student. Final answer: The investigation links a variable statistical question to a distributed random sample, consistent collection, distribution analysis and a qualified conclusion about Year 7 travel times. Check: Every conclusion refers to the sampled travel-time data, reports variation and avoids extending the result beyond the defined school population.

Example 2

Example 2 Evaluating a biased survey of physical activity. A student surveys 120 members of school sports clubs to estimate weekly activity for all 600 Year 7 students. Although the sample is large, club membership makes highly active students more likely to be included, so the sample is a convenience sample with selection bias. Improve the design by randomly selecting students from the full Year 7 roll, or by randomly sampling the same proportion from each home group, and follow up nonresponses consistently. Report the sample size, response rate, activity distribution and remaining limitations before generalising cautiously to the Year 7 population. Final answer: The sports-club survey cannot represent all Year 7 activity reliably, and a random sample from the full cohort with transparent nonresponse reporting is stronger. Check: The revised sampling frame gives every Year 7 student a known opportunity for selection rather than restricting selection to sports-club members.

Example 3

Example 3 Application problem 1 Problem: A school surveys only students in the library at lunchtime about preferred sports. Identify the sampling problem and propose a better method for estimating the whole school's preference. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: The sample is biased because library users may not represent the whole school. A random or stratified sample across year levels and locations would better represent the population. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: The sample is biased because library users may not represent the whole school. A random or stratified sample across year levels and locations would better represent the population. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: Design a feasible investigation of Year 7 sleep duration that states a statistical question, population, sample, variable, collection method and one limitation. Plan: Write one clause for each required component and make the reference night consistent for every sampled student. Work: Ask 'How many hours do students in our Year 7 cohort usually sleep on a school night?', define all Year 7 students as the population, randomly sample students from every home group, collect self-reported hours for the same specified night, analyse the distribution, and acknowledge that recall and self-reporting may be inaccurate. Interpret: The plan aligns every investigation component and identifies a realistic measurement limitation. Final answer: Ask 'How many hours do students in our Year 7 cohort usually sleep on a school night?', define all Year 7 students as the population, randomly sample students from every home group, collect self-reported hours for the same specified night, analyse the distribution, and acknowledge that recall and self-reporting may be inaccurate. Check: The plan aligns every investigation component and identifies a realistic measurement limitation.

Curriculum examplesCopied content

Content description: plan and conduct statistical investigations involving data for discrete and continuous numerical variables; analyse and interpret distributions of data and report findings in terms of shape and summary statistics.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. Which is a statistical question: 'How many minutes do Year 7 students at this school spend reading on a weekday?' or 'How many minutes did Amira read yesterday?', and why?

    Check answer

    Answer: The Year 7 question is statistical because it expects varying numerical responses from a group, while the question about Amira seeks one fixed value.

    Hint: Choose the question that would produce a distribution rather than one answer.

    Why: A statistical question anticipates variation across a population or repeated process.

  2. 2. In a survey of 80 randomly selected students from all 620 students at a school, identify the population and sample.

    Check answer

    Answer: The population is all 620 students at the school, and the sample is the 80 randomly selected students.

    Hint: Separate the group the conclusion concerns from the group actually surveyed.

    Why: The population is the target group and the sample is the subset that supplies data.

  3. 3. A study records each student's journey time to school in minutes; classify the variable as discrete or continuous and justify your answer.

    Check answer

    Answer: Journey time is continuous because time is measured on a scale, even if the recorded minutes are rounded.

    Hint: Ask whether a more precise clock could record values between whole minutes.

    Why: Measurement remains continuous when reported to limited precision.

  4. 4. A school surveys the first 100 students entering the library about weekly reading time; identify the likely sampling bias.

    Check answer

    Answer: The convenience sample is likely to over-represent students who use the library and may read more than students who do not enter it.

    Hint: Identify who is easy to reach and who has no chance to be selected.

    Why: The selection location systematically changes which students can enter the sample.

  5. 5. Put these investigation stages in a defensible order: analyse the distribution, define the question and population, collect data consistently, select a sample, clean the data, and communicate a conclusion.

    Check answer

    Answer: The order is define the question and population, select a sample, collect data consistently, clean the data, analyse the distribution, and communicate a conclusion.

    Hint: Begin with what you want to know and end with what the evidence allows you to say.

    Why: Each stage supplies trustworthy information needed by the next stage.

  6. 6. A random sample of 50 Year 7 students has median travel time 16 minutes and range 48 minutes; write a cautious contextual conclusion.

    Check answer

    Answer: For the sampled students, a typical travel time was about 16 minutes and travel times varied widely across a 48-minute range, but sampling uncertainty means the exact values may differ for the whole Year 7 cohort.

    Hint: Mention both median and range and include language that distinguishes sample from population.

    Why: The conclusion interprets centre and spread while keeping its claim within the evidence's scope.

  7. 7. A voluntary online survey receives 400 responses from 2,000 students; explain why the large response count does not remove possible bias.

    Check answer

    Answer: Students who chose to respond may differ systematically from nonresponders, so voluntary-response bias can remain even with 400 responses.

    Hint: Compare people who decided to participate with those who ignored the survey.

    Why: Sample size does not correct a selection process that over-represents motivated respondents.

  8. 8. Design a feasible investigation of Year 7 sleep duration that states a statistical question, population, sample, variable, collection method and one limitation.

    Check answer

    Answer: Ask 'How many hours do students in our Year 7 cohort usually sleep on a school night?', define all Year 7 students as the population, randomly sample students from every home group, collect self-reported hours for the same specified night, analyse the distribution, and acknowledge that recall and self-reporting may be inaccurate.

    Hint: Write one clause for each required component and make the reference night consistent for every sampled student.

    Why: The plan aligns every investigation component and identifies a realistic measurement limitation.

  9. 9. A survey finds that students reporting more exercise also report higher wellbeing; evaluate the claim 'exercise caused the higher wellbeing' and write a justified alternative conclusion.

    Check answer

    Answer: The survey shows an association but cannot establish causation because other variables and self-selection may affect both measures, so a justified conclusion is that higher reported exercise was associated with higher reported wellbeing in this sample.

    Hint: State exactly what the data move together with and identify what the study did not control or manipulate.

    Why: Observational survey evidence supports an association rather than an unqualified causal claim.

  10. 10. A school surveys only students in the library at lunchtime about preferred sports. Identify the sampling problem and propose a better method for estimating the whole school's preference.

    Check answer

    Answer: The sample is biased because library users may not represent the whole school. A random or stratified sample across year levels and locations would better represent the population.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: A very large convenience sample must represent the population well.

Correction: Increasing a biased sample's size does not repair the systematic exclusion or over-representation created by its selection method.

Common mistake: The sample and population are interchangeable because both refer to people in the study.

Correction: Data are observed from the sample and used cautiously as evidence about the larger population.

Common mistake: An association found in a survey proves that one variable caused the other.

Correction: An observational association can be influenced by other variables and does not by itself establish causation.

Curriculum alignmentStart here

We are learning to plan and conduct statistical investigations and communicate findings that are justified by the sampled data.

A statistical investigation begins with a question that anticipates variation, identifies the population of interest and defines a measurable numerical variable. The sample, measurement procedure and units must match that question so the collected data can actually address it.

A representative sample depends on selection method rather than size alone. Simple random or carefully implemented systematic sampling can reduce selection bias, while convenience and voluntary-response samples often over-represent accessible or strongly motivated people; nonresponse can introduce further bias.

After collection, data are checked, displayed and summarised by centre, spread and shape before a conclusion is written in context. The conclusion should distinguish the observed sample from the broader population, disclose limitations and avoid causal language unless the investigation design supports cause.

Success criteria

  • I can write a statistical question and define its population, sample and numerical variable.
  • I can select a feasible sampling and data-collection method while identifying possible bias and nonresponse.
  • I can analyse a distribution and write a contextual conclusion with appropriate limitations.
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