The idea, step by step
Three controls on interval width
A known-spread mean interval has width . Width doubles if population standard deviation doubles. It increases with the critical value. Quadrupling the sample size halves width.
These are comparisons under the same model. A new sample can have a new centre; a proportion interval can also have a new estimated spread. State which quantities are held fixed.
Plan precision without losing validity
To divide a margin by r, multiply the sample size by r squared. Reducing a margin from 3 to 1 needs nine times the observations at the same confidence and spread.
No increase in a convenience sample repairs omitted groups, leading wording or measurement bias. A larger sample may be more precise about the wrong population.
Check coverage with a computational simulation
Use a probability program or spreadsheet to draw 2,000 independent samples from a normal population with mean 100 and standard deviation 20. For each sample, calculate its mean, construct an interval and record whether the fixed value 100 is between its endpoints. Repeat with sample sizes 25 and 100, and confidence levels 95% and 99%.
The table gives a supplied reproducible simulation so you can interpret results without needing to run software. It used NumPy default_rng with seed 20261106, separate samples for the two sizes, and the same means for the two confidence comparisons at each size. Variation in capture counts is expected. Under the known-spread model, interval width is fixed for each size/confidence combination.
| Sample size | Confidence | Captures | Capture fraction | Mean width |
|---|---|---|---|---|
| 25 | 95% | 1903 | 0.9515 | 15.680 |
| 25 | 99% | 1978 | 0.9890 | 20.608 |
| 100 | 95% | 1902 | 0.9510 | 7.840 |
| 100 | 99% | 1988 | 0.9940 | 10.304 |
A common trap
Doubling n halves the margin.
Instead: It multiplies margin by 1/√2.
See the thinking
Half the margin
A margin is 4 at n = 50. Find n for margin 2 with spread/confidence fixed.
- Halving the margin requires four times the observations.
- 50 × 4 = 200.
Competing changes
Population SD doubles and n quadruples. Compare margins at the same confidence.
- Spread gives factor 2; sample size gives factor 1/2.
- Multiply the factors to obtain 1.
Check your understanding
Margin is 6 at n = 100. Find it at n = 900 with other quantities fixed.
Show answer
2.
Nine times the sample divides the margin by three.
Your turn
6 questions · 13 marks · Work at your own pace.
Read the questions online
- Margin is 5 at n = 36. Find it at n = 144.2 marks
- A sample of 80 has a given margin. How many observations give one-third that margin?2 marks
- SD changes from 8 to 12 with confidence and n fixed. Compare widths.2 marks
- Compare 99% and 95% widths using critical values 2.576 and 1.96.2 marks
- Both n and population SD are multiplied by four. Compare margins.2 marks
- Use the supplied repeated-interval simulation table. Compare interval width when n rises from 25 to 100 at fixed confidence. Does the observed capture fraction have to equal the advertised confidence exactly?3 marks
One simulation: 2,000 sample intervals per row; normal population mean 100, SD 20 Sample size Confidence Captures Capture fraction Mean width 25 95% 1903 0.9515 15.680 25 99% 1978 0.9890 20.608 100 95% 1902 0.9510 7.840 100 99% 1988 0.9940 10.304
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Margin is 5 at n = 36. Find it at n = 144.2 marks
Four times the sample halves the margin.
2.5. - A sample of 80 has a given margin. How many observations give one-third that margin?2 marks
Multiply by 3² = 9.
720. - SD changes from 8 to 12 with confidence and n fixed. Compare widths.2 marks
12/8 = 1.5.
The new width is 1.5 times the old width. - Compare 99% and 95% widths using critical values 2.576 and 1.96.2 marks
2.576/1.96 = 1.314285… .
Ratio approximately 1.3143. - Both n and population SD are multiplied by four. Compare margins.2 marks
Spread factor 4 divided by square-root sample factor 2 gives 2.
The new margin is twice the old. - Use the supplied repeated-interval simulation table. Compare interval width when n rises from 25 to 100 at fixed confidence. Does the observed capture fraction have to equal the advertised confidence exactly?3 marks
One simulation: 2,000 sample intervals per row; normal population mean 100, SD 20 Sample size Confidence Captures Capture fraction Mean width 25 95% 1903 0.9515 15.680 25 99% 1978 0.9890 20.608 100 95% 1902 0.9510 7.840 100 99% 1988 0.9940 10.304 Quadrupling sample size halves standard error and width under the known-spread model.
Each row contains a finite number of simulated intervals; capture fraction is an estimate of long-run coverage.
The normal population, fixed sample size and stated independent draws make the simulation method reproducible.
Width halves; no, empirical coverage varies randomly.
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 ↗