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Suppose you construct a 95% confidence interval on a mean. Can you state that there is a 97.5% probability that the true population mean lies below the CI upper bound and that there is a 97.5% probability that the true population mean lies above the CI lower bound?

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    $\begingroup$ This question may be helpful. $\endgroup$ Jul 18, 2014 at 15:26
  • $\begingroup$ There are actually two reasons why the answer is 'no' in general. The first is given in Alexis' answer and SeanEaster's link. The second reason is that $1-\alpha$ confidence intervals do not necessarily put exactly $\alpha/2$ in each tail. Indeed when dealing with a discrete parameter, a CI for it generally can't. $\endgroup$
    – Glen_b
    Jul 18, 2014 at 17:00

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No.

The confidence interval (specifically this beast: $\bar{x} \pm z_{(1-\text{CL})/2}\hat{\sigma}_{\mu}$, where CL is the confidence level) is not a statement about the probability of the mean, $\mu$, which either is or is not contained within any specific interval with no possibility of both conditions obtaining. The substantive meaning of the frequentist confidence interval does not really reduce beyond:

If you repeated the study that produced your sample data many, many times so that you gathered many many samples of size $N$ you would expect that the CL% confidence intervals constructed on the means of each of those samples would contain the true population mean CL% of the time.

Of course, there are other kinds of estimation and inference intervals with different substantive meanings.

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  • $\begingroup$ Suppose you have a set of data, compute the sample mean and the CL% confidence interval on the mean. What does the single CI that you have tell you, and what statement can you make with the single CI that you have? $\endgroup$ Jul 18, 2014 at 16:12
  • $\begingroup$ You can say you are this confident, where confident has the meaning I gave in my second paragraph. It's not especially pretty. :\ That said, it is a way of evaluating $\sigma_{\mu}$, which, if $\hat{\sigma}_{\mu} \approx \sigma_{\mu}$ is true, should give you an idea of the trustworthiness of $\bar{x}$ as an estimate of $\mu$... where "trustworthiness" is a synonym for "confidence." Sorry, it just doesn't get much better. $\endgroup$
    – Alexis
    Jul 18, 2014 at 16:19
  • $\begingroup$ OK thanks. So if I understand you correctly, you cannot say that there is a 95% chance that the single confidence interval you have contains the mean? $\endgroup$ Jul 18, 2014 at 16:22
  • $\begingroup$ That is correct. $\endgroup$
    – Alexis
    Jul 18, 2014 at 16:23
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    $\begingroup$ @Alexis Here you go: stats.stackexchange.com/questions/552783/… $\endgroup$ Nov 18, 2021 at 18:59

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