7: Confidence Intervals
- Page ID
- 58876
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Chapter 7: Confidence Intervals
Imagine you’re trying to estimate the average rent for a one-bedroom apartment in your city. You can’t look at every apartment listing in the entire market, so you take a sample say, 50 listings and calculate the average rent from just those. But here’s the big question:
How close is your sample mean to the real population mean?
This is where confidence intervals come in. Instead of giving a single number as an estimate, a confidence interval gives a range that likely contains the true population value. It’s one of the most important ideas in modern statistics, because it helps us make informed guesses while acknowledging uncertainty.
Key idea: A confidence interval is built around your sample statistic.
It takes the form: estimate ± margin of error
Confidence intervals pop up everywhere: opinion polls, clinical studies, economic estimates, even weather forecasts. Instead of saying “we know the exact truth,” we say something more honest (and useful): “We’re this confident that the truth lies somewhere in here.”
In this chapter, we’ll explore:
- How to use a sample to estimate a population parameter
- The structure and interpretation of a confidence interval
- How to find critical values (like \(z^*\) and \(t^*\))
- The difference between intervals for proportions and for means
- How sample size and confidence level affect the accuracy and width of intervals


