When Do You Use the Normal Distribution vs the T-Distribution for a Confidence Interval?

Use the normal distribution when the population standard deviation, $\sigma$, is known. Use the t-distribution when $\sigma$ is unknown and must be estimated using the sample standard deviation, $s$, regardless of sample size.

This is the required distinction in AHL 4.16: Confidence intervals.

Here, $\bar{x}$ is the sample mean, $n$ is the sample size, and $z^*$ or $t^*$ is the critical value for the chosen confidence level. The is or . A higher confidence level increases the critical value and therefore widens the interval.

The standard error measures the expected variation in sample means. When $\sigma$ is unknown, replacing it with $s$ introduces additional uncertainty, so the t-distribution has heavier tails and normally produces a wider interval.

For example, suppose $n=25$, $\bar{x}=72$, and the confidence level is $95\%$.

If $\sigma=10$ is known, $z^*=1.96$:

$$72 \pm 1.96\left(\frac{10}{\sqrt{25}}\right)=72\pm3.92$$

The interval is $(68.1,75.9)$ to three significant figures.

If instead only $s=10$ is known, use $t^*\approx2.064$ with $24$ degrees of freedom:

$$72 \pm 2.064\left(\frac{10}{\sqrt{25}}\right)=72\pm4.128$$

The interval is approximately $(67.9,76.1)$, which is wider.

Both intervals are centred at the same sample mean, but the t-interval is wider because estimating $\sigma$ adds uncertainty. As $n$ increases, the t-distribution approaches the normal distribution, so the difference between the critical values becomes smaller. This does not change the IB selection rule.

A common misconception is that a large sample automatically means using the normal distribution. Although the t-distribution approaches the normal distribution as $n$ increases, IB Mathematics: Applications and Interpretation HL requires the t-distribution whenever $\sigma$ is unknown.

Exam technique: Identify whether the question gives $\sigma$ or $s$, show the correct GDC confidence-interval setup, and interpret the interval in context. Do not say there is a stated probability that the fixed population mean lies inside the calculated interval.