# Does the Confidence Interval Contain a Mean If It Equals One of the Limits?

Does a confidence interval contain a mean if the proposed mean equals one of its limits?

For example, say I have a 95% confidence interval ranging from 3.19 to 3.49. Does this interval contain the mean 3.19?

• Intervals for discrete parameters would normally include their endpoints (because if you have an open interval with a given coverage, there's a shorter interval with the same coverage); for continuous quantities it's an event of probability zero, so it really doesn't matter whether it includes its endpoints.. Sep 28, 2018 at 3:20

If continuous variables are concerned (I guess yours is), it shouldn't really matter, because the probability of the RV being equal to an exact value is actually 0. You should consider the mechanics behind it to understand better. While calculating these intervals, we find some probabilities in the form $$P(X \leq u)$$ or $$P(X, which are the same. First one produces a closed CI, while the latter produces an open one. If you're that unlucky, i.e. your mean is exactly the same with your limit, you should first consider adding more resolution to your CI or mean calculation. Round it to more decimals than two.