I always thought that when comparing a statistic from two populations, say p_experiment and p_control, you take a pooled variance and create a confidence interval around 0. Then, if the statistic from p_experiment-p_control lies outside of this confidence interval, you conclude that the statistics in both populations are actually different.
However, I've been seeing examples of confidence intervals around the statistic of p_experiment-p_control. If this interval includes 0, we accept null, otherwise we reject the null.
When should I use one over the other? Or are both supposed to give the same answer? An example would be appreciated.
Would it make sense to take a confidence interval around p_experiment or p_control and see whether it includes the other statistic? Why or why not?