What is a Confidence Interval?
When we collect a sample from a population, we can calculate a sample mean . But this is just a point estimate — a single number that almost certainly isn't exactly equal to the true population mean . A confidence interval gives us a range of plausible values for , along with a stated level of confidence.
A confidence interval (CI) is an interval estimate of a population parameter, constructed from sample data, that is designed to contain the true parameter value with a specified probability (the confidence level).
The confidence level, denoted and usually expressed as a percentage (e.g., 90%, 95%, 99%), represents the proportion of all possible confidence intervals (constructed using the same method) that would contain the true population parameter.
A confidence interval is always written in the form:
where is called the margin of error. The margin of error depends on:
- The chosen confidence level
- The sample size
- The variability in the data ( or )
Think of a confidence interval like casting a net to catch a fish (the true mean). A 95% confidence level means your net-casting method catches the fish 95% of the time. Any single cast either catches the fish or it doesn't — but you trust the method.
The Two Scenarios: Known vs Unknown σ
The formula you use to construct a confidence interval for a population mean depends entirely on whether the population standard deviation is known or unknown.
| Situation | Distribution Used | Critical Value |
|---|---|---|
| known | Standard Normal () | |
| unknown | -distribution |
In IB AHL 4.16, the -distribution is used whenever is unknown — regardless of sample size. This is a key distinction from some older rules of thumb that switched to normal for large samples.
In nearly every real-world problem, you will not know , so you will almost always use the -distribution. If a question explicitly states the population standard deviation, use . If it gives you a sample standard deviation , use .