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# Investigating with samples

Data and chance · Samples

<!-- Speaker note: Students estimate a hidden population value from samples, then plan an ethical sample investigation and state an inference with its uncertainty (AC9M8ST04, nodes 1 and 2). About 45 minutes (Predict and Run 10 min · Investigate 10 min · Modify 12 min · Practise and Make 13 min (the investigation itself is a class task)). -->

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## Today

**We are learning to** a sample investigation asks a question about a population, plans a fair and ethical sample, collects and describes the data, then makes an inference about the population and says how sure it can be.

- I can plan an investigation with a fair, ethical sample.
- I can make an inference about a population from a sample.
- I can say how sure I am, and why.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need sampling methods, describing samples and sampling variation (the last three lessons), and the Year 7 investigation cycle. The one idea: an inference estimates the population that was sampled, with its uncertainty stated (size and method). The common mistake is inferring beyond the population (Zara's one school to all Australian teenagers). The board hides the school's value until Show the population, so students commit to an estimate first. Ethics: consent, anonymity, no harm, and neutral questions; school approval for real surveys. The full investigation (collecting real data) is a class task: the Make step plans it. Use a class sample of a short anonymous form on a non-sensitive topic. -->

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## Why it matters

> You can't see the whole school's answer, only your sample's. How close is your estimate, and how would you say so honestly?

<!-- Speaker note: In real investigations you only see your sample. An **inference** is a conclusion about the population drawn from a sample. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**A random sample of 40 students has a mean sleep of 8.8 hours. What can you say about all the students at the school?**

- Nothing at all
- Their mean is about 8.8 hours, give or take a little
- Their mean is exactly 8.8 hours

<!-- Speaker note: What happens: About 8.8 hours, give or take a little. A random sample of 40 gives a fair estimate, not the exact value. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what 'give or take' would mean here. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fdata%2Fsample-investigations%2Fstep-2&code=widget%3A+sample%0Ameasure%3A+sleep%0Amethod%3A+choose%0Aslider+n%3A+10%2C+100%2C+20%2C+10%0Apopulation%3A+hidden%0Aplot%3A+values%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/data/sample-investigations/?full=step-2#step-2)

<!-- Speaker note: A good inference: - is about the population the sample came from - gives an estimate, not an exact value ("about") - says how sure you can be, and why (the sample's size and how it was chosen) -->

<!-- Speaker note: Teacher: ask for the estimate before the reveal. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* About 8.8 hours, give or take a little. A random sample of 40 gives a fair estimate, not the exact value.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Ethics

Here is the routine for an inference.

$$\text{Random sample of } 50 \text{ Year 9s at our school}$$

1) $\text{Median screen time } 3 \text{ hours, range } 6 \text{ hours}$ — What the sample shows.
2) $\text{Year 9s at our school: about } 3 \text{ hours}$ — Infer about the population sampled only.
3) $\text{A sample of 50 can be a little off; more students would narrow it}$ — State the uncertainty.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write Random sample of 50 Year 9s at our school and say "The sample and how it was chosen." Line 1: write Median screen time 3 hours, range 6 hours and say "What the sample shows." Line 2: write Year 9s at our school: about 3 hours and say "Infer about the population sampled only." Line 3: write A sample of 50 can be a little off; more students would narrow it and say "State the uncertainty." -->

<!-- Speaker note: Teacher: ask Zara who her sample can speak for. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Ethics

**Which is the most important ethical step in a survey about students' sleep?**

- Teachers read the answers aloud
- Every student must answer
- Students agree to take part and their answers are kept anonymous
- Names are written on each form

* Answer: Students agree to take part and their answers are kept anonymous

<!-- Speaker note: Ethical data collection asks permission, keeps answers private, and does no harm. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Zara reports her investigation. Click the line where her inference goes too far.**

```
 1  Zara takes a random sample of 40 students at her school and asks about phone use after lights out.
 2  Her sample's median is 45 minutes, and the range is 120 minutes.
 3  She concludes that teenagers across Australia use their phones for about 45 minutes after lights out.
```

* Answer: line 3

<!-- Speaker note: Line 3: her sample came from one school, so she can infer about her school, not all Australian teenagers. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Saying how sure

**Which sentence states the uncertainty best?**

- From a random sample of 40, students at our school probably sleep about 8.8 hours; a bigger sample would give a closer estimate
- Students sleep exactly 8.8 hours
- We are certain students sleep 8.8 hours
- Students sleep 8.8 hours, which proves they are tired

* Answer: From a random sample of 40, students at our school probably sleep about 8.8 hours; a bigger sample would give a closer estimate

<!-- Speaker note: Name the sample, use 'about' or 'probably', and say how the estimate could improve. -->

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## Plan and improve



- **Order the stages:** 
- **A bigger sample:** 
- **Anonymous answers:** 
- **Which is more reliable:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/data/sample-investigations/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Ordering the stages, and seeing that a bigger sample tightens the estimate, prepares them to plan their own. Hints, in order: Take a sample of 20 and one of 80 or more before you reveal the population. For students who need more support: A five-stage plan template: question, sample, collect (ethically), display and describe, infer with uncertainty. For early finishers: Run your investigation twice with two different random samples. How different are the results, and what does that say about your uncertainty? The hard part, on purpose: Saying how sure you are without overclaiming. Ask: who exactly does this sample speak for, and how close might it be? When students get stuck: A student writes an exact value for the population. Ask: Did you measure everyone? Fix: Use 'about' or 'probably', and give the sample. / A student generalises to a bigger group. Ask: Who could have been chosen for this sample? Fix: Only that population: the school, the year, the club. / A student forgets ethics in the plan. Ask: How will people agree to take part, and stay anonymous? Fix: Add consent, anonymity and a neutral question to the plan. -->

<!-- Speaker note: Teacher: ask why the bigger sample's estimate is safer. -->

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## One school

**A random sample of 30 Year 10 students at one school finds most want a later start time. What can be inferred?**

- Nothing
- Most Year 10s at that school probably want a later start
- Most students in Australia want a later start
- Every Year 10 at the school wants a later start

* Answer: Most Year 10s at that school probably want a later start

<!-- Speaker note: Infer about the population sampled: Year 10 at that school. -->

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## A fair question

**Which question is fair to ask in an anonymous survey about exercise?**

- Why are you so lazy?
- Don't you think you should exercise more?
- What is your name and weight?
- How many minutes did you exercise yesterday?

* Answer: How many minutes did you exercise yesterday?

<!-- Speaker note: Neutral, specific and respectful questions give honest data. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## More certain

**How can an investigation make its inference more certain?**

- Round the answer
- Use a convenience sample
- Ask only friends
- Use a bigger random sample

* Answer: Use a bigger random sample

<!-- Speaker note: Bigger fair samples vary less, so the estimate is more certain. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: variation

**Review: two random samples of 25 from the same school give means of 8.6 and 9.0 hours. What explains the difference?**

- A measuring mistake
- Sampling variation
- One sample was biased
- The school changed

* Answer: Sampling variation

<!-- Speaker note: Random samples hold different students, so their means differ a little. -->

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## Make: plan your investigation

Choose a question about your school (sleep, screen time, travel to school, lunch). Write the plan as five stages. If your teacher agrees, carry it out with a class sample and write the report.

<!-- Speaker note: What this step is for: Planning a full sample investigation, with ethics and uncertainty, shows they can apply every part of the unit. Hints, in order: Name the population first, then the sampling method and size. / Include how you will ask permission and keep answers private, and how you will say how sure you are. -->

<!-- Speaker note: Teacher: ask how you will protect the people who answer. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**A random sample of 50 students at your school shows a median of 2 hours of screen time on school nights. Write an inference with its uncertainty.**

* Students at our school probably have about 2 hours of screen time on school nights, from a random sample of 50; another sample might give a slightly different value.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This completes AC9M8ST04 and Year 8 Statistics. Where it leads: Year 9 and 10 statistics (comparing samples, scatterplots, evaluating claims in the media). -->