QCE Mathematical Methods - Unit 4 - Interval estimates for proportions
Confidence Intervals for Proportions | QCE Mathematical Methods
Learn QCE Mathematical Methods confidence intervals for population proportions, including margin of error, confidence level and interpretation.
Updated 2026-05-18 - 4 min read
QCAA official coverage - Mathematical Methods 2025 v1.3
Exact syllabus points covered
- Understand the concept of an interval estimate for a parameter associated with a random variable.
- Understand and use the approximate confidence interval $(\hat{p}-z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\hat{p}+z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}})$ as an interval estimate for $p$, the population proportion, where $z$ is the appropriate quantile for the standard normal distribution.
- Understand and use the approximate margin of error, $z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.
- Understand and use the relationship between margin of error, level of confidence and sample size.
- Understand that there are variations in confidence intervals between samples and that most, but not all, confidence intervals contain $p$.
- Model and solve problems that involve interval estimates for proportions, with and without technology.
A point estimate gives one number, such as $\hat{p}=0.42$. A confidence interval gives a range of plausible values for the population proportion $p$.
The approximate confidence interval used in Methods is:
$ \left(\hat{p}-z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\ \hat{p}+z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right) $
Original Sylligence diagram for confidence interval proportions.
The interval is centred at $\hat{p}$. The lower and upper endpoints are equally far from $\hat{p}$:
$ \text{lower endpoint}=\hat{p}-E $
$ \text{upper endpoint}=\hat{p}+E $
where $E$ is the margin of error.
Margin of error
The margin of error is:
$ E=z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $
So the interval can be written as:
$ \hat{p}\pm E $
A larger confidence level uses a larger $z$ value, which makes the interval wider. A larger sample size makes the interval narrower.
Common confidence levels use these approximate standard-normal quantiles:
| Confidence level | Middle area | Tail area on each side | $z$ value | | --- | --- | --- | --- | | $90\%$ | $0.90$ | $0.05$ | $1.64$ | | $95\%$ | $0.95$ | $0.025$ | $1.96$ | | $99\%$ | $0.99$ | $0.005$ | $2.58$ |
The $z$ value is larger when you want more confidence, because the interval must reach further into the tails of the standard normal distribution.
Interpretation
The population proportion $p$ is fixed. The interval is random because it depends on the sample. So avoid saying "there is a 95% probability that $p$ is in this interval" after the interval has already been calculated.
A better interpretation is:
If this sampling process were repeated many times, about 95% of the intervals produced by this method would contain the true population proportion.
Two interpretations are usually needed:
- The calculated interval gives plausible values for $p$ based on this sample.
- The confidence level describes what would happen over many repeated samples using the same method.
For example, if a calculated interval is $(0.323,0.417)$, a clean context statement is: "Based on this random sample, a plausible range for the true proportion of students who study with flashcards is about $0.323$ to $0.417$." The long-run statement is separate: "If this method were repeated many times, about $95\%$ of such intervals would capture the true population proportion."
Worked example
The midpoint of the interval is still $\hat{p}=0.37$. The margin of error is about $0.047$, so the interval extends $0.047$ below and above the sample estimate.
Choosing sample size or confidence
The margin of error formula shows the tradeoffs:
$ E=z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $
If $n$ increases, the square-root term decreases. Doubling $n$ does not halve the margin of error; to roughly halve the margin of error, you need about four times the sample size.
If confidence level increases, $z$ increases. That makes the interval wider, even if $\hat{p}$ and $n$ stay the same.
Simulation idea
Imagine the true value is $p=0.40$. If you repeatedly take samples and build 95% confidence intervals, most intervals should cross $0.40$, but not all. That is the point of the confidence level.
In a simulation, each interval is random because each $\hat{p}$ is random. Some intervals lie mostly to the left of $p$, some to the right, and a few miss the true value altogether. That does not make the method broken; it is exactly what "about $95\%
quot; means.Assumptions and reasonableness
The approximate interval relies on a random sample and a sample size large enough for the normal approximation to be reasonable. If the sample is biased, the interval can look precise while still being centred in the wrong place. If the sample size is tiny or $\hat{p}$ is very close to $0$ or $1$, the approximation may be weak.