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The answer to this question discusses problems associated with calculating P-values for dependent observations. Let's say you have observations from two different groups that are dependent. You consider carrying out a t-test to compare means of the two groups. However, aware that the calculated P-values in a t-test will likely be inaccurate, you calculate means of each group and their 95% confidence intervals. Does calculating confidence intervals avoid problems associated with calculating P-values when observations are dependent?

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No. Since confidence intervals convey the same inferential information as $p$-values ($\mu_0 \in \mathrm{CI} \iff p_{H_0:\mu_0=\mu}\geq \alpha$), they also share the same difficulties dealing with dependence.

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