What can we conclude when the confidence interval for a regression line slope contains 0? If my confidence interval, say $(-.4, .7)$, for a regression line slope contains $0$, I am looking for the strongest conclusion that I can make from that. Can I conclude that "there is no statistical significance that there is a non-constant linear relationship between the two variables involved"?
My flawed reasoning and possible source of confusion: Even though $0$ is in this confidence interval, the set $\lbrace0\rbrace$ has measure $0$, so the probability of having slope $0$ should always be $0$ (regardless of whether or not $0$ is in the confidence interval).

I would phrase it as follows.

We can’t even pin down if the relationship is positive or negative.

This gets out of the issues surrounding the fact that we kind of know the parameter is not exactly zero, but it shows how we lack much information. (Think of someone telling you that your stock portfolio is either going to increase or decrease in value. I would not find that helpful!)
