Understand the idea.
Start with the claim
Translate a claim into a parameter for a specified population and time. Then inspect the frame, selection, response and measurement methods. Only after those checks should interval narrowness influence the conclusion: correct calculations alone do not justify generalising beyond the study design.
A proposed parameter value inside the interval is compatible with the interval procedure, but is not proved exactly correct. A value outside it is less compatible under the assumptions. Neither result gives certainty; sampling variation, model inadequacy and systematic bias still matter.
State limits the margin does not cover
Distinguish precision from accuracy. A usual margin describes sampling variation, while recall errors, dishonest answers, missing groups and selective reporting need separate evaluation. Also compare the size of a discrepancy with the practical decision; a tiny numerical difference may have little consequence.
For two groups, overlap of separate confidence intervals does not by itself prove equality or provide a formal analysis of their difference. The difference is another estimator with its own uncertainty model. Report the separate estimates without claiming an unperformed comparison.
See the method in action.
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”.
Worked solution
- 1
Identify the proposed parameter value.
The statement claims an exact population mean, rather than merely a plausible average.
- 2
Compare it with both interval bounds.
500 mL is contained in the reported 95% mean interval.
- 3
Explain what containment supports.
500 mL is compatible with this interval procedure, but other values inside the interval are compatible too, so exact equality is not proved.
- 4
Retain the study qualifications.
The conclusion depends on sampling, measurement and distribution assumptions; an interval does not independently verify those assumptions.
500 mL is compatible, not proved.
Narrow but biased
An app poll reports 62% support with a one-percentage-point margin and claims to describe all residents. What needs checking?
Worked solution
- 1
Read the estimate and margin on the same scale.
A one-percentage-point margin would give nominal bounds 0.61 and 0.63, but these numbers alone do not validate the population claim.
- 2
Compare the frame with all residents.
Check who can access the app, whether all relevant resident groups are covered, and whether the intended population and time are clearly specified.
- 3
Examine selection and response.
Find out whether users were randomly selected and whether respondents differ from non-respondents; voluntary participation can favour people with stronger views.
- 4
Evaluate the narrowness claim.
A large reply count can reduce a model-based sampling margin while leaving selection or response bias. The frame, selection, response process and target population must be checked before generalising to all residents.
The frame, selection, response process and target population.
Try it yourself.
Check your understanding
Does a 95% interval (0.46,0.54) establish a majority?
Show answer & reasoning
No.
A majority requires . The interval includes proportions below, equal to and above , so it does not establish a majority at the stated confidence level.
Keep practising
6 worksheet questions · 12 marks
Read the questions online
- A 95% mean interval is . Is target mean 23 compatible? Is it proved?2 marks
- A 95% proportion interval is . 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 . 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 . 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 ↗