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What is the clearest way to state that a 95% CI has a lower bound of A and an upper bound of B.

Most commonly, I see:


But this seems to imply that the CI is a vector of length two. If I were speaking very precisely, I would say that the 95%CI ranges from A to B.

Perhaps there is a more precise notation, for example:



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$\mathbb{P}(X \in [A,B]) = 0.95$ – user17762 May 22 '12 at 1:48
You want to be careful about what your probability is with respect to, though. You're often estimating some parameter $\theta$, so it is correct to say $\mathbb{P}(\theta \in [A,B]) = 0.95$ if the probability is with respect to the construction of the interval and not the particular interval constructed, as the probability that $\theta$ is in that fixed interval is $0$ or $1$. – John Engbers May 22 '12 at 1:55
@JohnEngbers If I understand correctly - it would be more clear to state that $\mathbb{P}(X|\textrm{model,data} \in [A,B]) = 0.95$? If so, would it be common that this conditioning is implied by context, e.g. in the preceeding text? – Abe May 22 '12 at 2:22
@Abe : I don't think your proposal is clearer than anything else. In particular, you haven't said what $X$ is! It seems to be something to which you're assigning a probability, so it's an event. What event do you intend it to be. And what can it possibly mean to say that the model and the date are in an particular interval? What you propose as "more clear" is clearly wrong even though it's completely opaque. – Michael Hardy May 22 '12 at 12:50
@MichaelHardy (it is data, not date) I included this information to satisfy the recommendation from JohnEngbers that I provide information about how the interval was constructed. $X$ is any model output parameter (e.g. slope, intercept). – Abe May 22 '12 at 14:59

The closed interval $\{x : A \le x \le B\}$ from $A$ to $B$ is an interval and is denoted $[A,B]$. Therefore it makes sense to say that the interval is equal to $[A,B]$.

The open interval $\{x : A < x < B\}$ from $A$ to $B$ is an interval and is denoted $(A,B)$. That looks even more like the standard notation for an ordered pair of numbers, but it's a quite different thing denoted by the same notation. Usually the context will make it clear which meaning is intended.

It does not make sense at all to say the confidence interval is in $[A,B]$, i.e. $\mathrm{CI} \in [A,B]$. That the interval is in the interval---i.e. is a member of itself---is false.

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