Would wearing a face mask prevent norovirus when around infected persons?

Not completely: Norovirus is exceedingly infectious. Just being in the room where someone with norovirus who vomits has some risk. Masks probably help, but not foolproof. Most important is careful hygiene: minimum contact, lots of hand washing, protective gloves if contact with infectious material (e.g. vomit, diaper changing of infected child). Even with all this, you may be infected.