The idea, step by step
Start with the claim
Identify the target population and time, sampling frame, parameter and method before looking at narrowness. Correct arithmetic does not justify generalising beyond the study design. A parameter inside an interval is compatible with that procedure, not proved exactly true.
If a proposed value lies outside an interval, describe evidence against compatibility under the assumptions rather than declaring impossible certainty. Model failure and bias remain possible explanations.
State limits the margin does not cover
A usual margin quantifies sampling variation, not measurement error, dishonest answers, incomplete frames or selective reporting. A statistically precise tiny difference may be practically irrelevant to the decision.
Overlapping intervals for two groups are not a formal test of equality. The difference needs its own inferential model. Describe the separate estimates honestly and avoid conclusions the analysis has not justified.
A common trap
Inside the interval means proved true.
Instead: It means compatible with the interval procedure, subject to assumptions.
See the thinking
Compatible does not mean proved
A 95% mean-fill interval is (497.8,501.2) mL. Evaluate “this proves the mean is exactly 500 mL”.
- 500 is inside the interval, but many other values are too.
- The conclusion also depends on the sampling, measurement and model assumptions.
Narrow but biased
An app poll reports 62% support with a one-percentage-point margin and claims to describe all residents. What needs checking?
- A large count can make a computed interval narrow.
- It does not ensure app users or volunteers represent all residents.
Check your understanding
Does a 95% interval (0.46,0.54) establish a majority?
Show answer
No.
It includes values on both sides of 0.5.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- A 95% mean interval is (21,25). Is target mean 23 compatible? Is it proved?2 marks
- A 95% proportion interval is (0.56,0.64). Give a cautious majority statement.2 marks
- A researcher reports only the narrowest interval from 20 post-hoc exclusions. What is wrong?2 marks
- Two group mean intervals overlap. Why is “the means are equal” unjustified?2 marks
- A mean difference is precisely estimated at 0.02 mm. What else matters in an engineering decision?2 marks
- A 2024 survey is used to claim a 2026 preference. Give a limitation.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A 95% mean interval is (21,25). Is target mean 23 compatible? Is it proved?2 marks
23 lies inside a range containing other plausible means.
Compatible, not proved. - A 95% proportion interval is (0.56,0.64). Give a cautious majority statement.2 marks
Every interval value is above 0.5; do not claim certainty.
It supports a population proportion above 0.5 under the study assumptions. - A researcher reports only the narrowest interval from 20 post-hoc exclusions. What is wrong?2 marks
The final selection is not the same as a pre-specified repeated procedure.
Selective analysis can undermine the advertised coverage. - Two group mean intervals overlap. Why is “the means are equal” unjustified?2 marks
Separate intervals estimate separate parameters, not their difference directly.
Overlap does not establish equality; a difference needs its own model. - A mean difference is precisely estimated at 0.02 mm. What else matters in an engineering decision?2 marks
Statistical precision is not practical importance.
Whether that difference matters relative to tolerances, risk and cost. - A 2024 survey is used to claim a 2026 preference. Give a limitation.2 marks
Define the original population and time; later generalisation needs justification.
The population preference may have changed.
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 ↗