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Say I have a 95% confidence interval of the sample mean. Does that mean there is a 95% probability that the population mean lies within that interval?

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    $\begingroup$ What do you mean 95% probability? For given data the probability might be different. The confidence interval is a statement about a specific procedure (to emphasis that it is not a statement about the data). Frequentist interpretation would be, that If you are making the experiment over and over again, and you apply the procedure to derive 95% confidence interval for a specific statistics (in your case it is the mean), than for 95% of the experiments the mean would be in the interval. $\endgroup$
    – them
    Commented Apr 25, 2016 at 12:44

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This is a common error in the interpretation of a confidence interval. Consider the explanation from the following site: http://onlinestatbook.com/2/estimation/confidence.html

"Confidence intervals for means are intervals constructed using a procedure that will contain the population mean a specified proportion of the time, typically either 95% or 99% of the time.

It is natural to interpret a 95% confidence interval as an interval with a 0.95 probability of containing the population mean. However, the proper interpretation is not that simple. One problem is that the computation of a confidence interval does not take into account any other information you might have about the value of the population mean. For example, if numerous prior studies had all found sample means above 110, it would not make sense to conclude that there is a 0.95 probability that the population mean is between 72.85 and 107.15. What about situations in which there is no prior information about the value of the population mean? Even here the interpretation is complex. The problem is that there can be more than one procedure that produces intervals that contain the population parameter 95% of the time. Which procedure produces the "true" 95% confidence interval? Although the various methods are equal from a purely mathematical point of view, the standard method of computing confidence intervals has two desirable properties: each interval is symmetric about the point estimate and each interval is contiguous."

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