I want to pose a question, mainly theoretical in nature, but applied responses are more than welcome. First, I'll define some terms.

An interval from a to b is said to be

An interval from a to b is said to be

So, that being said, are confidence intervals open or closed intervals? Feel free to include and sources and absolutely include your rationale! Try to clarify if the argument is practical vs theoretical.

I'll start with two arguments, one for each side. Please critique!

Open- that is, the CI (a,b) does not include it's endpoints.

1. Alpha is the

2. A (1-alpha)*100 CI is constructed by finding +/-Z(alpha/2) for the endpoints.

3. As noted before, these are also the minimum (in magnitude) values for which we Reject Ho.

4. Therefore, the CI must not include its endpoints, and is therefore an open interval. That is, for a test statistic equal to a or b we reject Ho.

Closed- that is, the CI [a,b] does include it's endpoints.

1. Acceptance region method wherein values exceeding (in magnitude) Z(alpha/2) lead to a rejection of Ho. Fail to Reject Ho occurs otherwise.

2. Therefore, the CI must include its endpoints, and is therefore a closed interval interval. That is, for a test statistic equal to a or b, we will fail to reject Ho.

I'm not being totally rigorous here, partly because I don't do math-type on the computer, but also because I'm sure many could run circles around me with more rigorous exposition, so I'll leave that to them.

Overall, I think a CI is an open interval!

An interval from a to b is said to be

**closed**if it is the case that for any value*x*in the interval from a to b, a**<=**x**<=**b. That is, the interval is inclusive of the stated endpoints and is denoted [a,b]. Values can take on a or b.An interval from a to b is said to be

**open**if it is the case that for any value*x*in the interval from a to b, a**<**x**<**b. That is, the interval is NOT inclusive of the stated endpoints and is denoted (a,b). Values can become increasingly close to a and b, but may never reach a or b.So, that being said, are confidence intervals open or closed intervals? Feel free to include and sources and absolutely include your rationale! Try to clarify if the argument is practical vs theoretical.

I'll start with two arguments, one for each side. Please critique!

Open- that is, the CI (a,b) does not include it's endpoints.

1. Alpha is the

**maximum**acceptable probability of a Type I error. Therefore, in a two tailed test, +/-Z (alpha/2) is the smallest (in magnitude) value Z statistic for which we will reject Ho. Fail to Reject Ho occurs otherwise.2. A (1-alpha)*100 CI is constructed by finding +/-Z(alpha/2) for the endpoints.

3. As noted before, these are also the minimum (in magnitude) values for which we Reject Ho.

4. Therefore, the CI must not include its endpoints, and is therefore an open interval. That is, for a test statistic equal to a or b we reject Ho.

Closed- that is, the CI [a,b] does include it's endpoints.

1. Acceptance region method wherein values exceeding (in magnitude) Z(alpha/2) lead to a rejection of Ho. Fail to Reject Ho occurs otherwise.

2. Therefore, the CI must include its endpoints, and is therefore a closed interval interval. That is, for a test statistic equal to a or b, we will fail to reject Ho.

I'm not being totally rigorous here, partly because I don't do math-type on the computer, but also because I'm sure many could run circles around me with more rigorous exposition, so I'll leave that to them.

Overall, I think a CI is an open interval!

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