Mojo,
I'm assuming when you say "shut your phone off" you mean turn the screen off, with the phone still "on" waiting for incoming calls.
When using the AA interface, yes it keeps playing and would keep playing even if you really fully power off the phone. The phone is just commanding the AA what to do. The AA is the server.
When using Pandora, yes if you turn the screen off because Pandora is built to keep streaming when the phone "sleeps". If you fully power the phone down, then Pandora will stop because in this case, the iPhone is acting as the server.
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