The idea, step by step
Fixed parameter, varying intervals
Under this frequentist model the population parameter is fixed. Before sampling, the endpoints are random. A correct 95% procedure captures the parameter in about 95% of repeated samples under its assumptions. An observed interval either contains it or does not.
“We are 95% confident” describes the reliability of the method. It does not turn the fixed mean into a random value with a measured 95% chance of lying inside the particular observed bounds.
Do not answer a different question
A confidence interval for a mean estimates the population average, not the range containing 95% of individuals. It also does not claim that 95% of future sample means lie inside this specific observed interval.
For the same data and model, higher confidence uses a larger critical value and a wider interval. A narrow biased interval can be confidently centred on the wrong target; good design is still essential.
A common trap
A 95% mean interval contains 95% of individual observations.
Instead: It estimates the population mean; individual spread is a different quantity.
See the thinking
A defensible interpretation
Interpret a 95% mean travel-time interval (48,52) minutes.
- Identify the mean as the parameter.
- The method captures the fixed mean in about 95% of repeated samples; it does not describe 95% of individual trip durations.
Expected coverage
A correct 90% interval method is repeated independently 200 times. How many captures are expected?
- Expected captures = 200 × 0.90.
- Expected counts are long-run averages, not fixed quotas.
Check your understanding
Is a 99% interval narrower than a 95% interval from identical data?
Show answer
No, it is wider.
Greater coverage requires a larger range.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Correct: “95% of residents have support rates between 0.52 and 0.60.”2 marks
- A 95% procedure is used 400 times independently. Find expected misses.2 marks
- Why can the realised count of misses differ from its expected count?2 marks
- Can a correctly calculated 95% interval miss the true mean?2 marks
- Explain the precision trade-off when confidence rises from 90% to 99%.2 marks
- Why is a confidence interval for a mean not a prediction interval for one new measurement?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Correct: “95% of residents have support rates between 0.52 and 0.60.”2 marks
A population proportion is one aggregate parameter, not a personal support rate.
We are 95% confident that the population support proportion is between 0.52 and 0.60. - A 95% procedure is used 400 times independently. Find expected misses.2 marks
400 × 0.05 = 20.
20. - Why can the realised count of misses differ from its expected count?2 marks
An expectation is an average over repetitions.
Repeated sample outcomes vary randomly. - Can a correctly calculated 95% interval miss the true mean?2 marks
The procedure misses in about 5% of repetitions under its model.
Yes. - Explain the precision trade-off when confidence rises from 90% to 99%.2 marks
The standard error stays fixed while the critical value increases.
The wider interval is less precise but has greater long-run coverage. - Why is a confidence interval for a mean not a prediction interval for one new measurement?2 marks
Mean uncertainty uses standard error, not the full individual spread.
Individual measurements vary more than sample means and need a different uncertainty model.
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 ↗