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.

I have seen SURMOUNT-2 (Lancet, 2023) cited in support of a claim I do not think it supports, twice this month, so I would like to work through what it actually shows.

My reading is that the trial is sound for its own question and is being stretched to answer a different one. I might be wrong about that, which is why this is a topic rather than a correction.

What I would like from this discussion: someone who disagrees with me to say why, with the section of the paper they are relying on.

This follows the opening post rather than contradicting it.

Confidence intervals: rather than a single point estimate, a range of plausible values. A narrow interval means precise measurement; a wide interval means measurement is imprecise. Wider intervals (more uncertainty) are honest about limitation.

Effect sizes: the magnitude of a difference, not just whether it is statistically significant. A difference that is significant (p<0.05) might be too small to matter. A large effect might not be significant if sample size is small.

Power and sample size: a study might be too small to detect a real effect (low power). Sample size calculations help determine how many participants are needed to detect an effect of a given magnitude.

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

Two things before anyone answers the substance.

First, the context in the first post is clear and specific. Second, the question is framed so that an answer can actually address it. Both are the norm here and both matter more than they sound.

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.

Absence of evidence and evidence of absence: if a study is small and finds no effect, that is absence of evidence, not evidence of absence. A larger study might find an effect that a small study missed.

P-values and significance: p<0.05 means the data would be surprising if the null hypothesis were true, not that the null hypothesis is false. A non-significant p-value does not mean "no effect".

Relative risk and odds ratios: both compare the rate in one group to the rate in another. Relative risk is easier to understand. Odds ratios are standard in many analyses but can be misinterpreted.

Thank you for the correction. I have edited my earlier post with a note rather than silently, so the thread still makes sense to read. The error was mine and it was the kind that comes from remembering a figure instead of looking it up.

This follows post #8 rather than contradicting it.

Confounding: a third variable explains an apparent association. In randomised data, randomisation balances confounders. In observational data, confounders can be adjusted for but unknown ones cannot.

I read post #10 twice before replying, because I had assumed the opposite.

Regression to the mean: if you select people with extreme values (very high or very low), their next measurement is often less extreme just by chance. This can look like a treatment effect when it is just statistics.

For anyone arriving from a search: the marked solution above is the direct answer, and the replies underneath it add the caveats that make it safe to use.