Definition: A confidence interval is a range of plausible values for a population parameter, built from sample data with a stated level of confidence such as 95 percent.
What is a confidence interval?
It has the form estimate ± margin of error. A 95% level means the method captures the true parameter in about 95% of samples. It does not say a given interval has a 95% chance of containing the parameter after it is computed.
How do you build one for a proportion?
Use p-hat ± z* times square root of p-hat(1 – p-hat)/n, with z* = 1.96 for 95%.
Worked example: 120 of 300 voters favor a measure, so p-hat = 0.40. The standard error is square root of (0.4 times 0.6/300) = square root of 0.0008, about 0.0283. The margin is 1.96 times 0.0283, about 0.0554. The interval is (0.345, 0.455).
Check conditions first: at least 10 successes and 10 failures, and a random sample.
What goes wrong?
Saying “95% of voters are in the interval.” The interval is about the parameter, not individuals. Mathbot words it as: we are 95% confident the true proportion lies between 34.5% and 45.5%.
Where to learn it
- Inference for Proportions: Intervals and Tests – lesson
- Inference for Means with t Procedures – lesson