Confidence intervals for dependent vs independent samples Suppose I have two independent samples, and I want to calculate a confidence interval for the difference between the population proportions.
Now suppose instead I have two dependent samples, and I want to do the same thing.
Which confidence interval would be wider, and why? Assume both samples are large enough for the Central Limit Theorem to come into effect.

For independent samples you could invert a Wald test to produce,
$$\bar{x}-\bar{y}\pm z_{1-\alpha/2}\sqrt{ \hat{\text{se}}_{\bar{X}}^2 + \hat{\text{se}}_{\bar{Y}}^2 }$$
For dependent samples you could invert a Wald test to produce,
$$\bar{x}-\bar{y}\pm z_{1-\alpha/2}\sqrt{ \hat{\text{se}}_{\bar{X}}^2 + \hat{\text{se}}_{\bar{Y}}^2 -2\hat{\text{cov}}_{\bar{X}\bar{Y}}}$$
If $\bar{X}$ and $\bar{Y}$ are positively correlated then the covariance term will be positive and the second interval will be shorter than the first.  If $\bar{X}$ and $\bar{Y}$ are negatively correlated then the covariance term will be negative and the second interval will be wider than the first.
