I realize this is a little late, but I have no idea where to start with building this device. How do you start overall with this event?
I would advise starting with learning circuits, and especially Ohm's Law. Electronics is the foundation for both the device and the written test, so this is step 1. Once you understand how resistors work, thermistors are only a small step up—they are resistors with constant resistance for a constant temperature, but who change resistance when the temperature changes.
Next you'll want to obtain a microcontroller, circuit pieces, a board, thermistors, and LEDs. If you can't buy them, ask your SciOly team or the physics teachers at your school to borrow some. It seemed like ages had to pass for me to make sense of microcontrollers (I used an Arduino), so I'd recommend using your microcontroller to experiment around with what you learned in step 1. See if you can build circuits where you can calculate the voltage across part of your circuit before actually measuring it, to test whether you've mastered electricity and your microcontroller board. Tutorials for microcontroller code are available online (although honestly a lot of the tutorials aren't all that great).
Finally, you'll want to make one of your thermistors waterproof, but keep the spare thermistors as backups, or for future SciOly students on your team. Then, as with any build event, you want to spend time testing your thermistor over and over in hot water and cold water. You should record the voltage measured for each temperature, being sure that you don't get multiple disparate voltages for a given temperature (that means something's broken). You'll then use these data to formulate an equation that allows you to convert from voltage to temperature. You can now adjust your device as needed to refine its temperature-gauging abilities. Good luck!
Admittedly, my way is the "long way" because each step only becomes clear after you've finished the previous. If anybody has a faster way or other insights, I'd love to hear about them!