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First of all, I have found this question: Interpretation of confidence interval which might seem as a duplicate.

As well as this explanation http://onlinestatbook.com/2/estimation/confidence.html within it -- but I find it hard to understand.

In the second link it is stated that:

Confidence intervals for means are intervals constructed using a procedure that will contain the population mean a specified proportion of the time, typically either 95% or 99% of the time. An example of a 95% confidence interval is shown below:

72.85 < μ < 107.15

There is good reason to believe that the population mean lies between these two bounds of 72.85 and 107.15 since 95% of the time confidence intervals contain the true mean.

But also:

It is natural to interpret a 95% confidence interval as an interval with a 0.95 probability of containing the population mean. However, the proper interpretation is not that simple. One problem is that the computation of a confidence interval does not take into account any other information you might have about the value of the population mean. For example, if numerous prior studies had all found sample means above 110, it would not make sense to conclude that there is a 0.95 probability that the population mean is between 72.85 and 107.15. What about situations in which there is no prior information about the value of the population mean? Even here the interpretation is complex. The problem is that there can be more than one procedure that produces intervals that contain the population parameter 95% of the time. Which procedure produces the "true" 95% confidence interval?

For the sake of the discussion I rather refer to a situation in which one can have huge amount of data, and the confidence level can be extremely high.

My questions are:

  • If one can't say it is as likely for the population parameter to be inside the interval as the confidence level, why does the author write "There is good reason to believe that the population mean lies between these two bounds (...)" in the first paragraph?

  • In the second quoted paragraph the author refers to a situation in which there was a prior information. I don't see how that is relevant. If your confidence level is, say $1-2^{-30}$, it is indeed very unlikely you get an interval which contradicts previous studies. If it indeed happens, one must conclude that one of you had a mistake. You, or the previous studies. Where am I wrong?

  • Also in the second paragraph the author writes: ... there can be more than one procedure that produces intervals that contain the population parameter 95% of the time. Which procedure produces the "true" 95% confidence interval? I didn't understand this line, what is he trying to say?

To summarize, I'll try to compare it to null hypothesis rejecting, which I understand better:

If I randomly pick a confidence interval from a set of confidence intervals, of which $1-2^{-30}$ contain the true population parameter, why can't I say it is as likely I have picked a good interval, as it is likely the null hypothesis should be rejected when $p\leq2^{-30}$?


Note: I am a beginning math student. I have taken with passion some basic math classes such as Linear Algebra 101, and Calculus, but nothing more. In statistics I have a reasonable understanding of basic hypothesis testing (null hypothesis, statistical significance, p-values).

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