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After calculating the 90% confidence interval for a mean I get that:

Lower Limit = 3
Mean = 4
Upper Limit = 7

How can I use this results given the following interpretation of CIs?

Were this procedure to be repeated on numerous samples, the fraction of calculated confidence intervals that encompass the true population parameter would tend toward 90%. Source: https://en.wikipedia.org/wiki/Confidence_interval

Where do the numbers 3 and 7 fit in the definition? Each of the samples result on different calculated confidence intervals. Why is the calculation of the CI necessary in the first place?

The following definition of confidence intervals uses the 3 and 7:

Interval that will bracket the true value of the parameter in 90% of the instances of an experiment that is repeated a large number of times Source: Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?

Is the first one just wrong?

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Where do the numbers 3 and 7 fit in the definition?

They don't - the confidence interval procedure referred to in the definition is the method used to calculated those two numbers (i.e., the confidence interval formula). Each new sample would yield new bounds for the interval. So you got bounds of 3 and 7 for the actual data, but if the procedure were to be repeated numerous times on new data, you would get new bounds each time. The definition is saying that if you repeated this over and over again, the bounds you get would encompass the true mean value 95% of the time (in the limit).

(I cannot find the second definition in the link you give. In any case, it does not appear to be correct. There is certainly nothing about a 95% confidence interval that would make it bracket the true parameter in 100% of instances.)

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    $\begingroup$ +1 The other key is that you do not know anything about whether the mean in this particular case actually falls within the Confidence Interval. It either does or does not, with 100% certainty, but we don’t know which. $\endgroup$ – Wayne Apr 2 at 23:19
  • $\begingroup$ Thanks for the answer. @Ben but then why is the calculation of the CI beneficial in the first place? My 3 and 7 don't help me have any confidence about the true parameter. Maybe are all the intervals going to have the same width? $\endgroup$ – italo Apr 3 at 12:40
  • $\begingroup$ Also when you say 'actual data' do you mean the population or my sample? Thanks again $\endgroup$ – italo Apr 3 at 12:43
  • $\begingroup$ @italo: The sample. $\endgroup$ – Reinstate Monica Apr 4 at 6:23
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    $\begingroup$ @italo: Perhaps this related answer will help explain the benefits. $\endgroup$ – Reinstate Monica Apr 4 at 6:23
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Let's say that you are interested in estimating the mean anxiety score in a population of patients (e.g., patients at a local hospital). You select 100 patients at random from this population and record their anxiety score. You compute the mean anxiety score for these patients and it comes out to be 4 (exactly the number you reported here). You also compute the 90% confidence interval for the mean anxiety score in the underlying population of patients and it comes out as 3 to 7 (which is what you reported here). How do you report this?

You can say things like:

Our sample of n=100 patients was observed to have a mean anxiety score of 4 points, with a 0.90 2-sided confidence interval ranging from 3 to 7. The data are consistent (within 90% confidence) with a mean anxiety score in the underlying population represented by these patients ranging from 3 to 7 points.

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