Noise Reduction Moving Average FIR
Since this is homework/a learning assignment, I think a hint is worth more than the (admittedly done on this site a couple of times) complete proof:

You forgot to also read the context: That's not true for just any noise and any signal, it's only true for noise that's white and a signal that fits perfectly into the filter shape.

Think about what "white" has to do with a moving average. Hope this helps.