# The interpretation of confidence interval

The confidence interval can be expressed in terms of a single sample: "There is a 90% probability that the calculated confidence interval from some future experiment encompasses the true value of the population parameter." Note this is a probability statement about the confidence interval, not the population parameter. This considers the probability associated with a confidence interval from a pre-experiment point of view, in the same context in which arguments for the random allocation of treatments to study items are made. Here the experimenter sets out the way in which they intend to calculate a confidence interval and to know, before they do the actual experiment, that the interval they will end up calculating has a particular chance of covering the true but unknown value.[4] This is very similar to the "repeated sample" interpretation above, except that it avoids relying on considering hypothetical repeats of a sampling procedure that may not be repeatable in any meaningful sense. See Neyman construction.

Suppose I have a 90% confidence interval for the weight of a population to be [100, 110]. So, how can I express it according the above interpretation?

It says about confidence interval from future experiments as if it completely ignores what I have already obtained, so how it covers the calculated interval.

Update

I think I understood it and just need confirmation: Because we have already obtained [100, 110] interval, this interval is fixed and no more a random variable, so we can say, we are 90% confident that this interval includes true mean.

However, if we regard confidence interval as a random variable, we can say there is a probability of 90% that 90% confidence intervals calculated this way include true mean...

right?

• It's a very interesting question and actually, you cannot interpret an obtained interval in that way: E.g. $\{\mu\in[100,110]\}$ is not an event with some probability, but it is true or false in a determinisic sense and you don't know which. So I think only the second statement is valid. It is easier in cases where your CI corresponds to a statistical test: Then the actual observation has a "meaning" in the sense that it is significant or not. Sep 18, 2020 at 11:28
• Sep 18, 2020 at 15:10

I confirmed my understanding at the same page. I said:

Because we have already obtained [100, 110] interval, this interval is fixed and no more a random variable, so we can say, we are 90% confident that this interval includes true mean.

However, if we regard confidence interval as a random variable, we can say there is a probability of 90% that 90% confidence intervals calculated this way include true mean...

I should also add that we can't either say there is a probability of 90% that true mean fall in [100, 110] because true mean is a constant and we use probability for random variables not constants...

A 95% confidence level does not mean that for a given realized interval there is a 95% probability that the population parameter lies within the interval (i.e., a 95% probability that the interval covers the population parameter).[13] According to the strict frequentist interpretation, once an interval is calculated, this interval either covers the parameter value or it does not; it is no longer a matter of probability. The 95% probability relates to the reliability of the estimation procedure, not to a specific calculated interval.

• yes, in the frequentist model. But we can treat the true mean as random as well as in Bayesian statistics.
– Dole
Sep 18, 2020 at 16:21
• @Dole Yes, but in the Bayesian case it would be called a credible interval not a confidence interval. Nov 5 at 3:23