Population estimates and uncertainty | Year 9

01 / LEARN

The idea, step by step

Scale the proportion, not the count

If a reasonably representative sample contains x successes out of n observations, use x/n to estimate the population proportion. Multiply this proportion by the population size. Do not multiply the raw success count by the population: the sample size is essential.

Use the word about or an approximate sign. You have measured only part of the population. If the result is 237.6 households, about 238 is sensible; claiming an exact fractional household is not.

estimated count = \frac{sample count}{sample size} \times population size

Samples vary

Different fair random samples can give different results. This is sampling variation, not automatically a calculation error. Repeated samples show how stable a method is. Larger fair samples usually fluctuate less, but a large biased sample can remain misleading.

Do not invent an exact margin of error from one small sample. Report the sample size, selection method and result, then explain limitations. Agreement between several samples strengthens the evidence without proving an exact census total.

Sample	Support out of 100	Estimate out of 500
1	52	260
2	58	290
3	55	275

A common trap

Multiply the number of sampled successes directly by the population.

Instead: First divide by sample size, then scale the resulting proportion.

02 / WORKED EXAMPLES

See the thinking

Example 1

Estimate a school count

In a random sample of 60 students, 36 usually bring lunch from home. Estimate the number among 900 students.

$36 / 60 = 0.6 = 60 %$ .
$0.6 \times 900 = 540$ .
This is a sample-based estimate, not a count of every student.

About 540 students.

Example 2

Read repeated estimates

Samples of 100 give 52%, 58% and 55% support. Estimate the corresponding counts in a population of 500.

Multiply each proportion by 500.
The estimates are 260, 290 and 275.
Their range is 30 people; different selected people can produce this variation.

260, 290 and 275; none is established as the exact population count.

Check your understanding

18 out of 30 sampled residents support a proposal. Estimate the count among 600 residents.

Show answer

About 360.

$18 / 30 = 0.6$ , then $0.6 \times 600 = 360$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A sample of 40 trees has 14 gum trees. Estimate the gum trees among 600 trees.
2 marks
52 of 80 surveyed students walk or cycle. Estimate the count among 500 students.
2 marks
A sample suggests 27% of 880 households own a dog. Give a sensible estimated count.
2 marks
Two fair samples give 42% and 47%. Must one be wrong?
2 marks
A biased sample of 1,000 gives 70%. A fair random sample of 150 gives 48%. Which method is preferable for the target population?
2 marks
Samples give 61%, 58%, 63% and 60%. Find their range in percentage points.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A sample of 40 trees has 14 gum trees. Estimate the gum trees among 600 trees.
2 marks
$14 / 40 = 0.35$ .
$0.35 \times 600 = 210$ .
About 210.
52 of 80 surveyed students walk or cycle. Estimate the count among 500 students.
2 marks
$52 / 80 = 0.65$ .
$0.65 \times 500 = 325$ .
About 325.
A sample suggests 27% of 880 households own a dog. Give a sensible estimated count.
2 marks
$0.27 \times 880 = 237.6$ .
Round to a whole household and retain about.
About 238 households.
Two fair samples give 42% and 47%. Must one be wrong?
2 marks
The selected people differ.
The sample proportions need not match exactly.
No. Ordinary sampling variation is possible.
A biased sample of 1,000 gives 70%. A fair random sample of 150 gives 48%. Which method is preferable for the target population?
2 marks
Representativeness matters.
Large size does not remove selection bias.
The fair sample of 150, assuming a suitable frame and response quality.
Samples give 61%, 58%, 63% and 60%. Find their range in percentage points.
2 marks
The largest is 63% and smallest 58%.
$63 - 58 = 5$ percentage points.
5 percentage points.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 9 sampling, comparative distributions, centre/spread/outliers, displays, critique and statistical investigations.

Official curriculum source ↗

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