Your question is Inferring Normals from Sums. Take a moment with it on the right.
Talk me through your thinking if you like. When you're confident, submit your answer and I'll grade it like a real screen (7/10 or better passes).
Explain how you infer normal random variables from observations of their sums.
State what can and cannot be inferred from the distribution of a sum alone. Describe the additional assumptions needed to infer normality of component variables, explain how you would assess normality empirically, and distinguish a theorem from a finite-sample diagnostic.