[samatman]

This is correct, which is why intervals don't choose an interpretation of the region of uncertainty.

If you do have reason to interpret the uncertainty as normally distributed, you can use that interpretation to narrow operations on two intervals based on your acceptable probability of being wrong.

But if the interval might represent, for example, an unknown but systematic bias, then this would be a mistake. You'd want to use other methods to determine that bias if you can, and correct for it.