Hi Terry,
You are absolutely right, the Hot Spot Analysis tool should be working on your selection set. In order to figure out exactly what the problem is, I'm hoping you can try a few things and answer a few questions to narrow it down.
1. When you use "Select by attributes" on the dataset, do the correct features appear selected in the map? If you open the attribute table with the selection do you see those features are selected? --If the answers to these questions aren't "yes" (ie the selection isn't working) it sounds like it is a problem with the selection, potentially with your query. Try the same thing with your definition query, too.
2. With the data selected, can you please try running another tool? Something simple, like maybe copy features or buffer. We want to make sure that your selection is being honored by other tools that are supposed to honor selections. If it isn't, then we may have an issue with selections and/or your dataset. If the selection is being honored, then would it be possible for you to share your data with us so that we can try to reproduce the issue and figure out what's going on?
So sorry you're having trouble, and hopefully we can get it figured out quickly!
Lauren Rosenshein
Geoprocessing Product Engineer