What does a 95% confidence interval actually tell you?
A 95% confidence interval is a range built by a method that captures the true population value in about 95% of random samples. Any single interval either contains the true value or it does not; the 95% describes the method, not the chance for one interval.
Estimate plus or minus a margin
Every confidence interval has the same shape: estimate ± margin of error, where the margin is a critical value times the standard error. For a population proportion, that is p̂ ± z* · √(p̂(1 − p̂) / n). The standard error measures how much the estimate would bounce around from sample to sample, and z* sets how many of those bounces you allow: 1.96 for 95%, 2.576 for 99%.
A worked interval
A random sample of 400 students at a college finds that 240, or 60%, study mainly the night before exams. The conditions hold: the sample is random, 400 is under 10% of the student body, and both 240 and 160 are at least 10.
The standard error is √(0.6 × 0.4 / 400) ≈ 0.0245. At 95%, the margin of error is 1.96 × 0.0245 ≈ 0.048, so the interval is 0.552 to 0.648.
Interpretation in context: we are 95% confident that the true proportion of all students at this college who study mainly the night before exams is between 55.2% and 64.8%. Raise the confidence to 99% and the margin grows to about 0.063, so the interval widens. Quadruple the sample size and the margin halves.
Misreadings to avoid
- "There is a 95% probability that the true proportion is in this interval." Once computed, the interval either contains it or not; the 95% is about the method.
- "95% of students fall in this range." The interval is about one population value, not individual data.
- "The sample proportion is probably in the interval." It is always in the interval, at the center.
- Thinking the margin of error covers bias. It covers random sampling variation only, not a badly worded question or a non-random sample.
Intervals and tests are linked: when a 95% interval for a mean leaves out a value, a two-sided test of that value at α = 0.05 will reject it. See p-values for the testing side.
Seeing 100 intervals at once
Ask Isotoon "What does a 95% confidence interval tell you?" and you get a short narrated animation. The clearest version draws 100 samples, builds an interval from each, and stacks them against a line at the true value: about 95 cross it and a few miss. The bell shape behind z* comes from the central limit theorem. More is on AP Statistics and statistics.
Questions
- Why does a higher confidence level give a wider interval?
- To catch the true value more often, the method has to cast a wider net. A 99% interval uses a larger critical value than a 95% interval, so its margin of error is bigger.
- How do I make a confidence interval narrower?
- Take a larger sample or accept a lower confidence level. Because the standard error shrinks with the square root of n, you need four times the sample to cut the margin in half.
- What is the correct way to interpret a confidence interval on the AP exam?
- State the confidence level, the parameter in context and the interval: we are 95% confident that the interval from A to B captures the true proportion (or mean) of the population described in the problem.