7. (1995) From your home outside Salt Lake City, Utah you’re off to visit the in-laws in Clearwater, Florida. You love the convenience of flying into St. Pete–Clearwater International Airport but one thing you want to avoid is flying through Chicago. O’Hare or Midway - it doesn’t matter. Be it weather of broken down airplanes, you’ve had nothing but bad luck in the Windy City. Thankfully, your travel agent has found you a convenient connection involving two nonstop flights, each of them operating the same aircraft type. Identify the two airlines, the aircraft type and the connection airport. (HINT: It’s not in Florida)
Depending on the time between flights, I believe one would indeed have had to carry on for this itinerary, as ATA didn't have any interline agreements for either ticketing or checking baggage 
DL had a bunch of SLC-XXX-ATL flights (XXX being large-ish midwestern airports such as OKC, MCI, MSP and the like), so the issue is where ATA had a footprint of more than one or two flights to IND/MDW ... how about Milwaukee/MKE
Milwaukee is an excellent guess (I recall ATA's Florida service out of there once being the subject of a question here) , so much so that I had to go double check just to be sure I didn't miss it. Alas, it doesn't work because although ATA has a nonstop MKE-PIE, it's a 700a departure that doesn't match up with Delta's inbound from SLC. So, we're looking for a different midwestern city.
BTW, the connection time on the itinerary I found for this question comes to 1 hour 55 minutes. If everything were on time and no bags were misplaced, it would be conceivable to check bags on both flights.