Can you interpret a confidence interval without knowing the population mean?

Question

If we don't know the population mean, could we still make a precise interpretation of the confidence interval? For example, would we still be able to say that there is a 90% chance that the population mean is in the confidence intervals or would we have to say that it cannot be interpreted?

Answer 1

Yes, you can interpret the CI to say that there is a 90% chance (assuming the CI used an alpha of 10%) that the population mean μ would be in the CI. This is one of the purposes of calculating a CI.

Also, if you are given a CI and the population distribution is assumed to be any symmetric distribution (normal, binomial, uniform) then you can also determine the sample mean (x bar) by finding the middle of the CI.

ex CI = (22.4 , 24.6) x bar = 22.4 + (24.6-22.4)/2 or 23.5

To be clear: This answer depends on what you mean by having a confidence interval. If you know the span (basically twice the margin of error) of the expected confidence interval because you know the standard error, you still can not say that you have a confidence interval because you only have the span. You do not have the upper bound, lower bound, or the mean.

If you have an actual interval you must have a lower bound and an upper bound. Then, if the distribution of the variable is any symmetric distribution the mean would be in the middle between the LB and UB (mean = (UB-LB)/2 + LB)