Margin of error and sample-size calculations | Year 12

01 / LEARN

The idea, step by step

Rearrange the margin inequality

For a mean with known spread require $n \geq {(z^{*} σ / E)}^{2}$ . For a proportion use a planning value $p_{0}$ and require $n \geq {(z^{*})}^{2} p_{0} (1 - p_{0}) / E^{2}$ . A margin stated in percentage points must first be expressed on the 0–1 proportion scale.

Without a defensible prior proportion, 0.5 is conservative because it maximises p(1−p). A planning value is not a known actual population proportion.

n \geq {(z^{*} σ / E)}^{2} or n \geq {(z^{*})}^{2} p_{0} (1 - p_{0}) / E^{2}

Round in the direction that meets the target

A sample-size formula gives a lower bound, not a nearest-integer estimate. If the bound is 384.16, use at least 385 completed valid observations. Round only after retaining calculation precision.

A response-rate estimate can guide the number of contacts to budget for. It does not guarantee completions or remove non-response bias. Check achieved sample size and response composition.

A common trap

Enter a margin of 3% as 3 in a proportion formula.

Instead: Enter 0.03.

02 / WORKED EXAMPLES

See the thinking

Example 1

Mean precision

Known SD is 12. Plan a 95% mean interval with margin at most 2.

$n \geq {(1.96 \times 12 / 2)}^{2} = 138.2976$ .
Round up.

139 valid independent observations.

Example 2

Proportion without prior estimate

Plan a 95% proportion interval with margin at most 0.05.

Use $p_{0} = 0.5$ : $n \geq {1.96}^{2} (0.25) / {0.05}^{2} = 384.16$ .
Round up.

385 valid observations.

Check your understanding

Use p₀ = 0.2, 95% confidence and margin 0.04. Find n.

Show answer

385.

${1.96}^{2} (0.2) (0.8) / {0.04}^{2} = 384.16$ ; round up.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Known SD is 15. Plan a 95% mean interval with margin at most 3.
2 marks
Without prior p, plan a 95% proportion interval with margin at most 0.04.
2 marks
For planning p₀ = 0.3, 95% confidence and margin 0.05, find n.
2 marks
A study needs 385 completions and expects a 70% response rate. Give a contact-budget estimate and limitation.
2 marks
Why is p₀ = 0.5 conservative?
2 marks
Why does rounding a bound of 120.02 down to 120 fail?
2 marks

Read the worked solutions

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

Known SD is 15. Plan a 95% mean interval with margin at most 3.
2 marks
${(1.96 \times 15 / 3)}^{2} = 96.04$ ; round up.
97.
Without prior p, plan a 95% proportion interval with margin at most 0.04.
2 marks
${1.96}^{2} (0.25) / {0.04}^{2} = 600.25$ ; round up.
601.
For planning p₀ = 0.3, 95% confidence and margin 0.05, find n.
2 marks
${1.96}^{2} (0.3) (0.7) / {0.05}^{2} = 322.6944$ ; round up.
323.
A study needs 385 completions and expects a 70% response rate. Give a contact-budget estimate and limitation.
2 marks
385/0.70 = 550.
Monitor actual response and non-response bias.
550 contacts; no guarantee of count or representativeness.
Why is p₀ = 0.5 conservative?
2 marks
The squared term is non-negative, so the maximum occurs at 0.5.
$p (1 - p) = 0.25 - {(p - 0.5)}^{2} \leq 0.25$ .
Why does rounding a bound of 120.02 down to 120 fail?
2 marks
The target margin inequality is not met below that bound.
120 is below the required lower bound; use 121.

Checking saved progress…

Continue learning

Assessing claims and limitations

Next lesson

Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods: sampling distributions and confidence intervals for means and proportions. Mean tasks distinguish known population spread from sufficiently large-sample estimated-spread normal intervals. Repeated-interval computational simulation, approximation conditions, design and interpretation are included.

Official curriculum source ↗

Back to Sampling and confidence intervals