How Do You Calculate a Confidence Interval for a Population Mean?

Calculate a confidence interval by taking the sample mean and adding and subtracting a margin of error. In IB Mathematics: Applications and Interpretation HL, use a normal interval when the population standard deviation is known and a t-interval when it is unknown.

The Method

A confidence interval has the form

$$\text{sample mean} \pm \text{critical value}\times\text{standard error}.$$

Here, $\bar{x}$ is the sample mean, $n$ is the sample size, and $z^*$ or $t^*$ is determined by the confidence level. Under AHL 4.16, use the normal distribution when is known and the t-distribution when is unknown, regardless of sample size.

The standard error measures how much the sample mean would vary between repeated random samples. It decreases as $n$ increases because division by $\sqrt{n}$ narrows the interval. A higher confidence level uses a larger critical value, so it produces a wider interval.

When $\sigma$ is unknown, enter the sample statistics $\bar{x}$, $s$, and $n$ into the GDC's one-sample t-interval routine. The calculator determines the appropriate $t^*$ using degrees of freedom. Do not replace with , and keep full precision until the final bounds.

For example, suppose $n=36$, $\bar{x}=52.4$, and the known population standard deviation is $\sigma=6$. For a 95% confidence interval, ${}^{}$:

$$52.4\pm1.96\left(\dfrac{6}{\sqrt{36}}\right)=52.4\pm1.96.$$

Therefore, the interval is

$$50.44\leq\mu\leq54.36,$$

or $(50.4,54.4)$ to 3 significant figures. In context: we are 95% confident that the population mean lies between $50.4$ and $54.4$.

A common misconception is that there is a 95% probability that the fixed population mean is inside this particular interval. More precisely, the method produces intervals containing $\mu$ in 95% of repeated samples.

Exam Technique

This is HL-only content, AHL 4.16, and every Math AI paper is GDC-active. Show the correct interval type and inputs before reporting both bounds, then interpret the interval in context; a calculator output alone may not earn all available marks.