Introduction to Confidence Intervals
Define a confidence interval as statistic ± margin of error. Interpret a 95% CI correctly: "We are 95% confident that the true parameter lies in this interval." Understand what confidence level means in terms of the long-run capture rate.
A lesson of AP Statistics, in Inference for Categorical Data: Proportions, about 10 minutes.
By the end of this lesson, you will be able to explain the logic of confidence intervals, correctly interpret a confidence level, and identify what does and does not affect interval width.
The Problem of Estimation
We want to know the value of a population parameter (like μ or p) but can only observe a sample statistic ($\bar{x}$ or $\hat{p}$). The statistic is an estimate — but how close is it to the truth?
A confidence interval gives a range of plausible values for the parameter, along with a measure of how confident we are that the true value falls in that range.
The General Form
$$\text{Statistic} \pm \text{Margin of Error}$$
$$= \text{Statistic} \pm (\text{Critical Value}) \times (\text{Standard Error})$$
Critical value ($z^$ or $t^$): reflects the confidence level; from the normal or t table.
Standard error: measures how much the statistic typically varies from sample to sample.
Margin of error: half the width of the interval; represents the maximum likely error.
Interpreting a Confidence Level
The confidence level (C%) tells you:
If we took many many random samples and constructed a C% confidence interval from each, approximately C% of those intervals would contain the true parameter.
This does NOT mean: "There is a C% probability that the true parameter falls in this specific interval."
Once an interval is constructed from a specific sample, the true parameter either IS or IS NOT in it — there's no probability left. The "C%" describes the procedure (the method), not a single interval.
The correct interpretation of a specific 95% CI (L, U): "We are 95% confident that the true [parameter] is between L and U [units]."
What Does the Confidence Level Mean Graphically?
Imagine 100 different samples, each producing a 95% CI. About 95 of those intervals will capture the true parameter; about 5 will not. You don't know which type your specific interval is — but the method produces correct intervals 95% of the time.
Factors That Affect Margin of Error
$$ME = z^* \cdot \frac{\sigma}{\sqrt{n}} \quad \text{(for a z interval)}$$