In a single study, a researcher can obtain a relevant summary statistic, and construct an $x$% confidence interval for that summary statistic.
Under the frequentist framework, this obtained confidence interval is thought of as being one of the infinitely many possible confidence intervals had the exact same study been Repeated infinitely many times.
Now think about a meta-analysis of 10 studies (i.e., loosely speaking, combining the results of 10 studies to make a summary of the 10 studies' summaries).
Again, here, one can also construct an $x$% confidence interval for the summary of the 10 studies' summaries (i.e., meta-analytic summary).
I was wondering how should one think of/interpret the Confidence Interval for the meta-analytic summary under the Frequentist Framework?
Precisely, There are two difficult notions that make interpreting the CI for the meta-analytic summary in my example difficult for me:
First, is the number of studies. I have 10 studies, and these are the only number of studies available on the topic of interest to me. So, how the number of studies affect the interpretation of a CI for a meta-analytic summary?
Second, is the number of repetitions. I know we must think of repeating our meta-analysis with 10 studies infinitely many times. But, when we have only 10 studies ever available what does infinitely many repetitions means, here (I mean there are 10 studies and they can only be meta-analyzed only one time)?