Confidence Intervals for the Mean (AI HL)
A confidence interval for the mean gives you a range that is very likely to contain the true population mean, built from a single sample. The tricky part is not the arithmetic - your GDC does that - it's choosing the right method and reading the output correctly. This page focuses on that choice, with worked examples and the mistakes that lose marks. It's part of the broader Confidence Intervals topic.
11 questions on this sub-topic.
Choosing your interval
Covered under IB syllabus reference AHL4.16: confidence intervals for the mean of a normal population, using the normal distribution when \(\sigma\) is known and the \(t\)-distribution when \(\sigma\) is unknown - regardless of sample size. You should also be able to interpret what the resulting interval means in context.
z-interval vs t-interval
\(\bar{x} \pm z^* \dfrac{\sigma}{\sqrt{n}}\) or \(\bar{x} \pm t^* \dfrac{s}{\sqrt{n}}\)
These aren't given as algebra in the formula booklet - your GDC builds them directly from raw or summary data, so the real skill is spotting which routine to use. If the question states the population standard deviation \(\sigma\), use a \(z\)-interval; if you only have a sample standard deviation \(s\) (however large \(n\) is), use a \(t\)-interval.
Confidence level
90% → 95% → 99%
A higher confidence level needs a larger critical value, which widens the interval.
Need the full syllabus wording and GDC walkthrough? See Confidence Intervals.
Worked examples
A sample of \(n=25\) has mean \(\bar{x}=50\) and standard deviation \(s=4.\) Find a 95% confidence interval for the population mean.
Worked solution
Use a \(t\)-interval with \(n=25\); technology gives M1
\((48.3,\ 51.7).\) A1
\((48.3,\ 51.7).\) A1
A population has known standard deviation \(\sigma = 10\). A sample of \(n = 64\) gives \(\bar{x} = 75\). Construct a 95% CI using the \(z\)-interval formula \(\bar{x} \pm z^* \dfrac{\sigma}{\sqrt{n}}\), where \(z^* = 1.96.\)
Worked solution
\(\text{SE} = \dfrac{10}{\sqrt{64}}\) M1
\(= 1.25.\) A1
Margin \(= 1.96 \times 1.25\) M1
\(= 2.45.\) A1
CI: \((72.55,\ 77.45).\) A1
Common mistakes
- Reaching for \(z\) by default. Unless the question explicitly gives you the population standard deviation \(\sigma\), you should be using a \(t\)-interval with \(s\) - it doesn't matter how large the sample is.
- Assuming a higher confidence level is always "better". A 99% interval is wider, not more accurate - it trades precision for a greater chance of containing the true mean, which is worth stating explicitly if a question asks you to compare levels.
- Losing the units or context in the final answer. A confidence interval is only fully answered once you've stated it in context - "the mean mass lies between 20.6 g and 23.1 g" - not just as a bare pair of numbers.
Ready to practise properly?
10 confidence-interval questions on the mean, marked instantly like the real exam.
Quick answers
When do I use a z-interval instead of a t-interval for the mean?
Use a \(z\)-interval only when the population standard deviation \(\sigma\) is given directly. Use a \(t\)-interval whenever you only have the sample standard deviation \(s\), regardless of how large the sample is.
Why does a 99% confidence interval come out wider than a 95% one?
A higher confidence level needs a larger critical value, which widens the margin of error and therefore the interval - you are trading precision for a higher chance of capturing the true mean.