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Why it's said when we estimate a parameter, from 100 confidence intervals, 95 will have the parameter. And why it is incorrect to say that is a 95% probability that the true parameter is in the confidence interval? Thank you in advance.

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    $\begingroup$ You generate the confidence interval once. Suppose it is constructed to be a 95% confidence interval. The true value of the parameter is unknown but fixed. So the actual interval may include it or it might not. The interval was constructed in such a way that if we repeated the process a large number of times approximately 95% of the intervals will contain the true parameter value and the rest will not. For the given interval we cannot assess a probability because the parameter is a fixed quantity and not random. $\endgroup$ Commented Apr 24, 2017 at 21:23
  • $\begingroup$ @Alexis There has been so much discussion about this issue on this site that I was not sure it was important enough to make it an answer. But if you and the OP think I should make it an answer I will do that. $\endgroup$ Commented Apr 24, 2017 at 22:25
  • $\begingroup$ As gung mentions there is a lot of interesting information in the link he gave. $\endgroup$ Commented Apr 25, 2017 at 0:26
  • $\begingroup$ Although some of the answers in the link go beyond confidence intervals for a mean this question and my answer are more general. The concept goes back to Neyman and is different from Fisher's concept of fiducial inference. $\endgroup$ Commented Apr 25, 2017 at 0:52
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    $\begingroup$ I explained why the question is not an exact duplicate of the referenced post. I see that the OP has used the self-study tag. I am not quite sure why it requires that tag which I am very much acquainted with. $\endgroup$ Commented Apr 25, 2017 at 17:38

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The confidence interval is only calculated once. For example, if you construct a 95% confidence interval the interpretation is that in repeated sampling, the interval will include the true parameter approximately in 95% of the cases. You do not actually generate multiple intervals because you have just one sample. It is the procedure that works 95% of the time. So some intervals (approximately 5%) will not contain the true parameter.

For the actual interval, you generate just one confidence interval and the true parameter may or may not fall inside. The parameter is fixed and not random so it does not make sense to say that the probability is 0.95 that the interval contains the parameter. That kind of interpretation can be made with Bayesian credible intervals but not with confidence intervals which are a frequentist construct.

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