How does one go about proving the characteristic of a confidence interval that:

A 95% confidence interval means if you were to randomly sample the same way 1000 times and create 1000 confidence intervals, approximately 95% of these intervals would contain the true parameter.

A confidence interval is calculated using the sample mean and the standard error so each confidence interval will be different (unique to the sample), and people talk about how to interpret the confidence interval (as above), but how does one go about proving that indeed 95% of these intervals would contain the true parameter/mean?


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    $\begingroup$ If the interval had the correct (95%) coverage, you would not generally get 950 in 1000 intervals containing the parameter -- in fact you'd only see 950 about 6% of the time, and you could easily get below 940 or above 960. $\endgroup$ – Glen_b Mar 10 '19 at 10:53

Confidence intervals are derived by the use of a pivotal quantity, which is a function of the data and the parameter of interest, that has a distribution that does not depend on any of the model parameters. To derive the confidence interval, one starts by making a probability statement about the pivotal quantity (usually that it falls within some bounds with a set probability) and then re-arranges the statement to make it a statement about the parameter of interest falling within some (random) bounds with that same fixed probability. To see how this is done for the standard confidence interval for a mean, see the derivation here. A general discussion of the method can also be found in O'Neill and Puza 2006.

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