QCE Specialist Mathematics - Unit 4 - Statistical inference

Confidence Intervals for Means | QCE Specialist Mathematics

Learn QCE Specialist confidence intervals for population means, margin of error, confidence level and interpretation.

Updated 2026-05-18 - 5 min read

QCAA official coverage - Specialist Mathematics 2025 v1.4

Exact syllabus points covered

  1. Understand the concept of an interval estimate for a parameter associated with a random variable.
  2. Understand and use the approximate confidence interval $\left(\bar x-z\frac{s}{\sqrt n},\bar x+z\frac{s}{\sqrt n}\right)$ as an interval estimate for $\mu$, the population mean, where $z$ is the appropriate quantile for the standard normal distribution.
  3. Understand and use the approximate margin of error: $E=z\frac{s}{\sqrt n}$.
  4. Understand and use the relationship between margin of error, level of confidence and sample size.
  5. Understand and use the concept that there are variations in confidence intervals between samples and that most but not all confidence intervals contain $\mu$.
  6. Use $\bar x$ and $s$ to estimate $\mu$ and $\sigma$, to obtain approximate intervals covering desired proportions of values of a normal random variable and compare with an approximate confidence interval for $\mu$.
  7. Model and solve problems that involve interval estimates for sample means, with and without technology.

A point estimate gives one number. A confidence interval gives a plausible range. For a population mean $\mu$, the sample mean $\bar x$ is the centre of the interval and the margin of error decides how far the interval extends.

Confidence intervals for means

Original Sylligence diagram for specialist confidence mean.

Confidence intervals for means

The formula

For large samples, the QCE Specialist approximate confidence interval for $\mu$ is:

$ \left( \bar x-z\frac{s}{\sqrt n}, \bar x+z\frac{s}{\sqrt n} \right). $

The margin of error is:

$ E=z\frac{s}{\sqrt n}. $

So the interval can be written as:

$ \bar x\pm E. $

If the population standard deviation $\sigma$ is known, the more direct normal interval is:

$ \bar x\pm z\frac{\sigma}{\sqrt n}. $

In practice, $\sigma$ is usually unknown, so the Specialist large-sample interval replaces $\sigma$ with the sample standard deviation $s$. That replacement is why the syllabus stresses large samples.

Choosing the $z$ value

For a two-sided confidence interval, the $z$ value leaves equal tail areas on both sides.

Common values are:

  • $z=1.645$ for a $90\%$ confidence interval
  • $z=1.96$ for a $95\%$ confidence interval
  • $z=2.33$ for a $98\%$ confidence interval
  • $z=2.58$ for a $99\%$ confidence interval

Higher confidence needs a larger $z$ value, so the interval becomes wider.

The $z$ value is chosen from the standard normal distribution. For a $95\%$ interval, $2.5\%$ is left in each tail, so the central area is $0.95$ and the critical value is about $1.96$.

Another way to think about the critical value is by upper cumulative probability:

  • $90\%$ confidence uses the $95$th percentile of $N(0,1)$
  • $95\%$ confidence uses the $97.5$th percentile
  • $98\%$ confidence uses the $99$th percentile
  • $99\%$ confidence uses the $99.5$th percentile

This is because the missing probability is split evenly between the two tails.

Interpretation

A $95\%$ confidence interval does not mean there is a $95\%$ probability that this one fixed interval contains $\mu$. The population mean is fixed. The interval changes from sample to sample.

A better interpretation is: if the same sampling process were repeated many times, about $95\%$ of the constructed intervals would contain the true mean $\mu$.

After the interval has been calculated, it either contains $\mu$ or it does not. The confidence level describes the long-run success rate of the method, not the probability of this fixed interval.

How changes affect the interval

Increasing the sample size narrows the interval because $\sqrt n$ is in the denominator.

Increasing the sample standard deviation widens the interval because the data are more variable.

Increasing the confidence level widens the interval because the $z$ value increases.

The full width is:

$ 2E=2z\frac{s}{\sqrt n}. $

Changing $\bar x$ shifts the interval left or right but does not change its width. Changing $s$, $n$ or the confidence level changes the width.

Estimating $\mu$ and $\sigma$

In practice, $\bar x$ estimates $\mu$ and $s$ estimates $\sigma$. The confidence interval uses both:

$ \left( \bar x-z\frac{s}{\sqrt n}, \bar x+z\frac{s}{\sqrt n} \right). $

Normal-value intervals are different. An interval such as $\bar x\pm 2s$ describes a rough range for individual values in a normal population. A confidence interval for $\mu$ describes uncertainty about the population mean. The first uses $s$; the second uses the standard error $\frac{s}{\sqrt n}$.

Planning sample size

Because:

$ E=z\frac{s}{\sqrt n}, $

larger $n$ reduces the margin of error. To halve the margin of error, you need about four times the sample size, assuming the same confidence level and similar standard deviation.

Simulation and data collection

A confidence-interval simulation works like this:

  1. Choose a population with known mean $\mu$.
  2. Take many simulated samples of size $n$.
  3. Build a confidence interval from each sample.
  4. Count how many intervals contain $\mu$.

For a correct $95\%$ method, the long-run capture rate should be close to $95\%$. Individual intervals still either capture $\mu$ or miss it.

The sampling process matters as much as the formula. If you want to estimate average height in a school, sampling only Year 12 students may bias the result. If you want a political estimate, sampling only one party's supporters is not neutral. More data narrows intervals only when the data are collected in a way that still represents the population.

Quick check

Sources