What does a 95% confidence interval actually assert?

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Answer

$That the procedure generating it captures the true parameter in 95% of hypothetical repetitions of the study.$
$The confidence is a property of the method across repetitions, not of the particular interval you computed, which either contains the parameter or does not.$
$This is why saying there is a 95% probability that the parameter lies inside a specific interval is a frequentist error, even though it is the interpretation everyone wants; a Bayesian credible interval is the object that supports that statement, and it requires a prior.$
$In practice the useful reading is about precision: a narrow interval near zero says the effect is small, which is informative, while a wide interval crossing zero says the study could not distinguish a meaningful effect from nothing, which is not the same as evidence of no effect.$

Check yourself — multiple choice

There is a 95% probability the parameter is in this interval
The procedure covers the true parameter in 95% of repetitions — coverage is a property of the method, not of the one interval; a credible interval is what supports the probability statement
95% of the data lies in the interval
The estimate is 95% accurate

Confidence describes long-run coverage of the procedure; probability statements about the parameter need a prior.

#confidence-interval#estimation

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