Opening
People regularly ask whether there is a 95-percent probability that an unknown mean lies inside one particular frequentist confidence interval. It is an understandable question, and it is generally the wrong interpretation.
Teaching narrative
A frequentist confidence interval is produced by a procedure with long-run coverage under a specified model. Before collecting data, the interval endpoints are random. The population parameter being estimated is fixed.
When the population standard deviation is known and observations are normal, a normal critical value gives an exact interval for a mean. When that standard deviation is estimated from the sample, the small-sample t distribution acknowledges the extra uncertainty.
Proportions need their own attention. The simplest Wald formula behaves badly near zero and one and with small samples. Wilson's score interval often provides a more trustworthy approximation under a binomial model.
None of these mathematical refinements cures survey nonresponse, a bad sampling frame, dependence, or measurements that answer the wrong research question. Before using a nominal interval in a survey, verify sampling frames, missingness, and independence assumptions as carefully as you verify the algebra.
Before the experiment
Must exactly 95 of 100 independently constructed 95% confidence intervals contain the true mean?
Live demonstration
Run the fifty-interval coverage laboratory at 90%, 95%, and 99%, then compare a t interval for five fictional observations with a Wilson interval for four successes in twenty trials.
Exit question
Explain the difference between a random procedure's coverage and a probability assigned to a fixed parameter.