IB Maths AI 4.11 Notes

Confidence intervals

A confidence interval gives a range of plausible values for a population mean. Instead of using a sample mean as a single estimate, we build an interval around it to show the uncertainty caused by sampling.

• If a confidence interval is narrow, the estimate is more precise.
• If it is wide, the estimate is less precise.

The general form is:

$\text{estimate} \pm \text{margin of error}$

For confidence intervals for a population mean, the estimate is the sample mean $\bar{x}$.

A confidence interval does not mean that the population mean definitely lies in the interval. Instead, it means that the method used is expected to capture the true population mean in a certain percentage of repeated samples.

For example, a $95\%$ confidence interval means that if the same method were repeated many times, about $95\%$ of the intervals produced would contain the true population mean.