Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have read two books that explicitly state that the $(1-\alpha)$% confidence interval should be interpreted as:

If you construct 100 such confidence intervals, $\alpha$ of them are expected to not contain the true population statistic and $(1-\alpha)$ of them are expected to contain the true population statistic.

and not as

There is a $(1-\alpha)$% probability that the true population statistic is contained in the confidence interval.

In my view, they both equivalent: If you make the first statement, you implicitly make the second statement. You are looking at any one arbitrary confidence interval, which in itself is a random variable, the generic confidence interval should, therefore, be subject to the second statement. Do things change when this random variable is actually realized?

share|improve this question
add comment

2 Answers 2

up vote 0 down vote accepted

I struggled with this concept for quite a while, but an intuitive explanation for it I found on Wikipedia (as background context a $95\%$ confidence interval of the number of voters voting for a particular party was found to be $[36\%,44\%])$:

For example, in the poll example outlined in the introduction, to be 95% confident that the actual number of voters intending to vote for the party in question is between 36% to 44%, should not be interpreted in the common-sense interpretation that there is a 95% probability that the actual number of voters intending to vote for the party in question is between 36% to 44%. This would be technically incorrect. The actual meaning of confidence levels and confidence intervals is rather more subtle. In the above case, a correct interpretation would be as follows: If the polling were repeated a large number of times (you could produce a 95% confidence interval for your polling confidence interval), each time generating about a 95% confidence interval from the poll sample, then approximately 95% of the generated intervals would contain the true percentage of voters who intend to vote for the given party. Each time the polling is repeated, a different confidence interval is produced; hence, it is not possible to make absolute statements about probabilities for any one given interval.

share|improve this answer
add comment

I think confidence intervals are better understood in the following way. There is an infinite set of constant open intervals. Some of them cover the population statistic $\theta$ and some of them don't. Now the (random) confidence interval is just a mechanism to draw an interval from this set. So the $(1-\alpha)\%$ is the chance of picking a constant interval that covers $\theta$. But whatever interval is drawn (or realized), since each of these interval is constant, the probability that it contains $\theta$ can only be $0$ or $1$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.