AP Statistics Confidence Intervals: What Are You Confident About?

To interpret a confidence interval, state the confidence level, the endpoints, and the population quantity being estimated. For a proportion, explain how confident you are that the interval captures the true proportion of a named population with a specified response. The confidence level describes the method’s performance over repeated random samples.

College Board’s AP Statistics Course and Exam Description, effective Fall 2026, covers constructing and interpreting these intervals in topics 3.3 and 3.4. The survey and practice questions below are original, fictional examples.

One survey, one interval, one population

A school has $2{,}000$ students enrolled this term. A simple random sample of $120$ students is asked, “Would you use the school library on Saturday if it opened?” All selected students answer, and $72$ say yes. Assume the responses accurately reflect how those students would answer this question.

The target is the proportion of all students enrolled at this school this term who would answer yes. It describes reported interest; actual Saturday attendance could differ.

The sample proportion is $72\div120=0.60$. Using a one-sample z-interval with $95\%$ confidence gives the following calculation:

$$0.60\pm1.96\sqrt{\frac{0.60(1-0.60)}{120}}$$

The margin of error is approximately $0.08765$. The endpoints, rounded to three decimal places, are $(0.512,0.688)$, or $51.2\%$ to $68.8\%$.

This follows the proportion-interval method explained in OpenStax’s lesson on a population proportion. Keep unrounded values through the calculation, then round the reported endpoints.

Before using that method, check the conditions in College Board’s topic 3.3: the sample is random; $120\le0.10(2{,}000)=200$, satisfying the condition for sampling without replacement; and the observed counts, $72$ yes and $48$ no, are both at least $10$.

Write the interpretation in context

A complete interpretation: We are $95\%$ confident that the interval from $51.2\%$ to $68.8\%$ captures the true proportion of all students enrolled at this school this term who would answer yes to the Saturday library question.

A reusable sentence is: “We are [confidence level] confident that [lower endpoint] to [upper endpoint] captures the true [population quantity] for [population and context].”

Fill in the quantity carefully. A proportion concerns a share with a particular response; a mean concerns an average measurement and needs units. Naming only “the students” leaves the quantity unclear.

What repeated sampling means

Imagine taking many new simple random samples of $120$ students from this same population. Compute a $95\%$ interval from every sample using the same procedure, with its conditions satisfied. In the long run, about $95\%$ of those intervals would capture the same true population proportion.

The sample results and endpoints vary. The population proportion stays fixed in this thought experiment. Once a particular interval has been calculated, it either captures that value or misses it; the sample does not tell us which occurred.

The $95\%$ level is therefore a long-run capture rate, not a guarantee that exactly $95$ of the next $100$ intervals will succeed. NIST’s explanation of confidence limits makes the same distinction for a population mean.

Four statements to repair

Use the interval to assess a claim

A claim that $60\%$ of the school’s students would answer yes is compatible with this interval because $0.60$ is inside it. That does not establish that the population proportion is exactly $0.60$.

A claim of $75\%$ is outside the interval. The interval provides evidence against that value under the stated method and assumptions. Because the entire interval is above $0.50$, it also supports a majority answering yes, while retaining sampling uncertainty.

Keep the conclusion within the sampled population. This survey does not establish interest at other schools, actual attendance, or a causal effect of opening the library. The random sampling and random assignment guide explains how study design controls which conclusions are justified.

Confidence does not fix a biased survey

For the same data and interval method, raising confidence from $95\%$ to $99\%$ uses a larger critical value and produces a wider interval. It does not improve the way the sample was selected.

If the school instead surveyed only students already visiting the library, the numerical interval would not account for that selection bias. Similarly, a large margin of error does not automatically cover nonresponse or misleading question wording. Check the study design before trusting a polished interpretation.

Try three interpretation checks

For these original exercises, assume an appropriate sampling design and interval procedure with conditions satisfied. The intervals are supplied so you can concentrate on their meaning.

1. A $90\%$ confidence interval for the proportion of all members of a particular robotics club who prefer morning meetings is $(0.31,0.47)$. Write a complete interpretation.
2. A $95\%$ confidence interval for the mean waiting time of all visitors to a particular school office during one week is $(6.4,8.0)$ minutes. Explain why “$95\%$ of those visitors waited between $6.4$ and $8.0$ minutes” is unsupported.
3. In the library example, another student says the $95\%$ level means that $95\%$ of students would answer yes. Correct the statement and name the sample’s actual yes percentage.

Check your next explanation

Underline the confidence level, endpoints, population, and quantity in your sentence. Then ask whether you accidentally described the sample, individual observations, or a different population. If so, record that specific error in the practice error-log template and rewrite the sentence before attempting a fresh question.

Try a free AP Statistics practice question with Kibo to continue with introductory statistics.

Prepared with AI assistance. Original calculations were checked separately; no human review is claimed. Sources checked October 25, 2026. Kibo is not affiliated with or endorsed by College Board.