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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.

Harvey Motulsky
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James C
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    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. – John Madden Aug 31 '22 at 23:26
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    Any single interval must either contain the estimated or not contain the estimated once constructed. 1 is the de jure interpretation. – Demetri Pananos Aug 31 '22 at 23:35
  • So the second is an incorrect interpretation of confidence interval, right? – James C Aug 31 '22 at 23:37
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    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? – Graham Bornholt Sep 01 '22 at 02:15
  • I looks to me that in both cases one should be constructing prediction intervals, because this is about expectations. – Robert Jones Sep 01 '22 at 06:16

1 Answers1

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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.

Harvey Motulsky
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