What a confidence interval means, from scratch — the long version

The question in the title: What a confidence interval means, from scratch — the long version I will give what I have already checked below so nobody repeats it.

Asking about confidence interval directly, because I have read four threads on it and each answered a slightly different question.

The version I want answered is the narrow one: given the method stated below, is the result within what anyone else has seen? I am not asking what it means yet.

Method, numbers and the two assumptions I am aware of making are below. If the assumptions are wrong that is more useful to me than agreement.

The opening post answers the question as asked. The question underneath it is different.

A confidence interval is a statement about a procedure rather than about this particular interval, and the practical reading — plausible values consistent with the data — is close enough for most purposes and worth knowing is an approximation.

Adding it because I spent an afternoon working it out and nobody should have to twice.

Two things can be true about confidence interval at once: the mechanism is plausible and the evidence for the size of the effect is thin. Most of the argument here is people defending the first against attacks on the second.

That is a fair summary of where the discussion has got to.

Coming back to post #3, because the follow-up matters more than the original answer.

Medians and means diverge for skewed distributions, and most of the quantities discussed here are skewed. Which one a paper reports is a choice worth noticing.

Correct me on the arithmetic if it is wrong; I would rather know.

Confidence interval is one of those subjects where the general answer and the answer for a specific case diverge, and the thread will go in circles until someone says which one is being asked for.

Practical note on confidence interval: write down what you expect before you look. The number of times I have found what I went looking for is higher than chance would allow.

Survivorship in a self-reporting population biases every aggregate produced from it, and the bias is in the flattering direction.

If anyone can point at the primary source I would be grateful.

Source for the confidence interval figure, since it was asked for. It is in the discussion rather than the abstract, which is why the version circulating is stronger than the paper is.

Reading the surrounding paragraph is worth the two minutes. The authors are more careful than their summarisers.

A request rather than an answer: could whoever has the primary source for confidence interval post it? I have seen the claim three times this month and each version had lost a qualifier.

Building on post #8 rather than restating it.

Number needed to treat: how many people need to be treated to prevent one bad outcome or achieve one good outcome. More intuitive than relative risk reduction.

Post #10 put the caveat in the right place and I want to underline it.

Confidence interval sits at the boundary between what this community can usefully discuss and what it cannot, and I think it falls on the discussable side, narrowly.

An honest declaration on confidence interval: I have a prior here and it is strong enough that you should weight what I say downward. Stating it rather than hiding it.