I have found discordant information on the question: "If one constructs a 95% confidence interval (CI) of a difference in means or a difference in proportions, are all values within the CI equally likely? Or, is the point estimate the most likely, with values near the "tails" of the CI less likely than those in the middle of the CI?
For instance, if a randomized clinical trial report states that the relative risk of mortality with a particular treatment is 1.06 (95% CI 0.96 to 1.18), is the likelihood of 0.96 being the correct value the same as 1.06?
I found many references to this concept online, but the following two examples reflect the uncertainty therein:
Lisa Sullivan's module about Confidence Intervals states:
The confidence intervals for the difference in means provide a range of likely values for ($μ_1-μ_2$). It is important to note that all values in the confidence interval are equally likely estimates of the true value of ($μ_1-μ_2$).
This blogpost, titled Within the Margin of Error, states:
What I have in mind is misunderstanding about “margin of error” that treats all points within the confidence interval as equally likely, as if the central limit theorem implied a bounded uniform distribution instead of a t distribution. [...]
The thing that talk about “margin of error” misses is that possibilities that are close to the point estimate are much more likely than possibilities that are at the edge of the margin".
These seem contradictory, so which is correct?