What is a good method for producing confidence intervals when computing a mean over hierarchical data? Specifically, I have the text messages sent by group of people and I want to compute the mean message length for people in the group with 95% confidence intervals. The number of messages sent by each person varies from 10 to 100. For computing the mean, I first compute the mean message length for each texter and then take the mean of these values over all texters. But I'm less sure how to compute good confidence intervals. Some ideas I've had based on bootstrap resampling:
- Bootstrap resampling the messages for each texter, keeping the set of texters fixed. This seems like it will by inaccurate if there is high variance in message lengths over different texters.
- Bootstrap resampling the texters, keeping the messages sent by each texter fixed. This seems like it won't capture that we're more confident about a texter's mean when that texter has sent 100 messages rather than 10.
- Doing both: first bootstrap resampling the texters, then bootstrap resampling the messages for each tester.
Any thoughts on which method is the most appropriate or if there is another technique I could use?