Confidence intervals for a mean: VCE Specialist Mathematics Unit 4

What this dot point is asking

VCAA wants you to construct an approximate confidence interval for a population mean from a sample, choosing the right $z$ value for the confidence level, computing the interval, understanding how sample size affects its width, and interpreting it correctly. The interpretation, in particular, is a frequent exam focus.

The confidence interval formula

An approximate $C\%$ confidence interval for the population mean $\mu$, based on a sample of size $n$ with sample mean $\bar{x}$, is

where $\frac{\sigma}{\sqrt{n}}$ is the standard error and $z$ is the value cutting off the central $C\%$ of the standard normal distribution. The quantity $z\,\frac{\sigma}{\sqrt{n}}$ is the margin of error. When the population standard deviation $\sigma$ is unknown, the sample standard deviation is used in its place for large samples.

Choosing the $z$ value

The $z$ value depends only on the confidence level:

• $90\%$ confidence: $z \approx 1.645$;
• $95\%$ confidence: $z \approx 1.96$;
• $99\%$ confidence: $z \approx 2.576$.

A higher confidence level needs a larger $z$, hence a wider interval, since more confidence requires casting a wider net.

The effect of sample size

The interval half-width is $z\,\frac{\sigma}{\sqrt{n}}$, so it shrinks as $\frac{1}{\sqrt{n}}$. A larger sample gives a narrower, more precise interval. To halve the width you must quadruple the sample size, the same diminishing return seen with the standard error.

Interpreting a confidence interval correctly

The confidence level describes the procedure, not a single interval. The correct statement is: if many samples were taken and a $C\%$ interval computed from each, about $C\%$ of those intervals would contain the true mean $\mu$. It is incorrect to say there is a $C\%$ probability that $\mu$ lies in a particular computed interval, because $\mu$ is a fixed (unknown) constant, and the computed interval either contains it or does not.

Examples in context

Example 1. A $90\%$ interval is narrower than a $99\%$ interval from the same data, because $1.645 < 2.576$.

Example 2. Increasing $n$ from $25$ to $100$ halves the margin of error, since $\sqrt{100} = 2\sqrt{25}$.

Try this

Q1. State the $z$ value for a $99\%$ confidence interval. [1 mark]

• Cue. $2.576$.

Q2. Find the margin of error for $\sigma = 15$, $n = 25$, at $95\%$ confidence. [2 marks]

• Cue. $1.96 \times \frac{15}{5} = 1.96\times 3 = 5.88$.

Q3. Give the correct interpretation of a $95\%$ confidence interval. [2 marks]

• Cue. In repeated sampling, about $95\%$ of such intervals would contain the true mean.