The work of Fisher (1959) and Buehler (1959) discuss the importance of conditioning on recognizable subsets of the sample space. The stopping time yields an easily identifiable partition of the sample space when considering group sequential testing. We first present confidence intervals that are correct when conditioning on the subset of data for which a trial stopped at a particular analysis. These intervals have very desirable properties for observations that are highly unusual (given any value of the mean). In addition, they provide insight into how information about the mean is distributed between the two sufficient statistics. We then use conditional coverage probabilities to compare the sample mean, stagewise, and repeated confidence intervals. We find that none of these intervals outperforms the others when conditioning on stopping time, and no interval is a uniformly acceptable performer.