Interpreting an Interval for a Mean

A confidence interval for a mean estimates one number: the true average of the whole population, often written mu. That true mean is fixed, we just don't know it. The interval from a sample gives a plausible range for it. The correct sentence template is: we are C% confident that the interval from [lower] to [upper] captures the true mean of [population, in context]. Notice this talks about the average, never about a single individual. An interval for average battery life tells you nothing about how long any one specific battery lasts, because individual values spread out much more than sample means do.

The C% confidence level describes the process, not any single result. Imagine taking many samples of the same size and building an interval from each one. About C% of those intervals would capture the true mean, and about (100-C)% would miss it entirely. Once you've actually computed one interval, like 44.2 to 51.9 hours, that interval either contains the true mean or it doesn't; there's no more randomness left to assign a probability to. So you cannot say there's a 95% chance the true mean is in this specific interval. The 95% is a statement about the reliability of the method you used to generate it, built up before you ever saw the data.

This interpretation step sits right after building the interval using a t-distribution, and it's where most wording mistakes cost understanding. The same logic you learn here carries forward unchanged when you interpret intervals for differences between two means later in the unit.