Say I have two populations, and I am computing a sample mean and a corresponding (1-$\alpha)100 \%$ confidence interval for each. Now, I understand that if the confidence intervals are disjoint, then we have a statistically significant difference. I also understand that if the confidence intervals are overlapping, then we cannot infer anything about the populations one way or another unless further analysis is done.

But what if the second point estimate is an element in the confidence set of the first point estimate? That is, what if we have $\bar{x}_2 \in C_{\alpha, \bar{x}_1}$, where $C_{\alpha, \bar{x}_1}$ is the confidence interval corresponding to the first point estimate, $\bar{x}_1$?

My impression is that this is not statistically significant, as now $\bar{x}_2$ is a possible value for the true population mean $\mu_1$, but since we are dealing with estimates (and thus standard errors), I am not so sure. More generally, is there a "degree of overlap" required before we can conclude "not statistically significant"?


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