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Usually, people tend to confuse the interpretation of confidence intervals. The following two statements are said to be quite confusing (even to myself):

  • A 95% Confidence Interval means that the population parameter will be contained within this interval with a probability of 0.95 (incorrect)
  • A 95% Confidence Interval means that the Confidence Interval itself has a 0.95 probability of containing the population parameter (less incorrect)

I was wondering if it is possible to create some example (e.g., an R simulation) which shows that why one of the above statements is more correct than the other.

I found this post here The importance of a correct interpretation of a confidence interval in which R code is provided that simulates data to illustrate these concepts.

But is it possible to create an example in which it becomes clear that interpreting a confidence interval as "the probability of the population parameter being contained within this interval" is clearly incorrect or results in a contradiction?

Perhaps someone can think of such an example or provide a link/reference to such an example?

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    $\begingroup$ A clearly incorrect example: $\mu = 10$, $\bar{x}=12$, and 95% CI is $(10.2, 13.8)$. $Pr\left[(10.2 < \mu) \cap (\mu < 13.8)\right] = 0$, despite 95% confidence level. $\endgroup$
    – Alexis
    Dec 26, 2022 at 2:31
  • $\begingroup$ While this can be an interesting question and a potentially useful one for teaching the green horns, one must tread carefully with "incorrect" interpretations: they are incorrect. Casella and Berger elaborated and beautifully explained the conundrum and the actual correct interpretation, which is briefly stated here. $\endgroup$
    – User1865345
    Dec 26, 2022 at 2:50
  • $\begingroup$ Seems to be like a philosophical point which would be difficult to establish with a sim study. $\endgroup$
    – John Madden
    Dec 26, 2022 at 3:36
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    $\begingroup$ We have several very highly voted threads addressing the subtleties of confidence intervals with many answers posted to each: see this site search $\endgroup$
    – whuber ♦
    Dec 26, 2022 at 14:44
  • $\begingroup$ Aside: There is a relatively straightforward and useful interpretation of frequentist confidence intervals: This interval estimates a plausible range of values for the true parameter at a given level of confidence. This deemphasizes the meaning of the Type I error rate (i.e. the 'reproducing the study a large number of times…' bit) without dispensing with it entirely. $\endgroup$
    – Alexis
    Dec 26, 2022 at 18:23

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