I understand that pre-data collection we can be 95% confident that the interval we're about to calculate will contain the true population parameter θ.
This doesn't mean that once we've calculated the C.I. that there's a 95% chance that θ is in the C.I.
θ is either contained within the interval or it isn't. As now both θ and the C.I. are fixed quantities
However, if we wished to calculate an interval with a 95% probability that that interval contained θ we could use a Bayesian credible interval.
I've seen this discussion in many places. These discussions also tend to include how often the misinterpretation of C.I. appear in the research literature and how few researches understand the correct interpretation when polled, or quizzed on the topic.
What I've failed to find is the real-world consequences of interpreting a C.I. as if it were a credible interval.
Can someone please point me in the direction of real-world consequences? Preferably a case-study, but even an example would be greatly appreciated.