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  1. A definition that I am aware of is interpreted as follows. Given a fixed probability parameter $\theta$, if I construct $100$ confidence intervals $\{I_1, \cdots, I_{100}\}$, then it is expected that $95$ of confidence intervals contain $\theta$.

  2. However, I came across an alternative interpretation that confuses me:

Suppose we are extracting 100 samples from a group of students in a university, where each sample has a certain number of records. We have calculated the mean age of students from each sample. Now, if we say that the confidence interval is [18, 24] with a 95% probability, then it means that the mean age of 95 out of 100 samples will lie in the range of 18-24.

I asked my friend, who is a stat PhD student, he said the second interpretation is more widely used in practice (unfortunately, I did not have enough time to catch details). However, I am unsure why 1. and 2. would imply the same thing: 1. is about constructing 100-many intervals and observing how many of these intervals contain the true parameter; meanwhile 2. is about fixing one interval, and count how many of 100 samples will have its respective parameter falling within the given interval. Any clarification would be appreciated.

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    $\begingroup$ They may have been referring to the fact that this is a prevalent misconception among nonstatistician practioners of statistics, rather than asserting this to be the correct interpretation. Though I do recall arguing with a PhD student friend who held that view :) And yeah No 1 is right: a 95% CI is one that contains the truth 95% of the time. $\endgroup$ Commented Aug 31, 2022 at 23:26
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    $\begingroup$ Any single interval must either contain the estimated or not contain the estimated once constructed. 1 is the de jure interpretation. $\endgroup$ Commented Aug 31, 2022 at 23:35
  • $\begingroup$ So the second is an incorrect interpretation of confidence interval, right? $\endgroup$
    – James C
    Commented Aug 31, 2022 at 23:37
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    $\begingroup$ As @Harvey and the others have written, number 2 is definitely incorrect. Interest is in whether the interval covers the parameter not whether future random intervals would fall inside this one. Was the second extract from a text? $\endgroup$ Commented Sep 1, 2022 at 2:15
  • $\begingroup$ I looks to me that in both cases one should be constructing prediction intervals, because this is about expectations. $\endgroup$ Commented Sep 1, 2022 at 6:16

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As @John and @Demitri said in comments, the first definition is correct and the second one isn't. Think of it this way: The selected sample may be very unrepresentative by chance, so the 95%CI computed from that sample may only include a small fraction of the means of the other samples.

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