If you did not change anything about the intake, installing a wider exhaust could be either neutral or detrimental. Many people have bought magic exhaust mods for their naturally-aspirated cars only to discover that the engine now guzzles fuel without being noticeably more powerful.

Of course, Cobb are well known for complete kits with a new intake manifold and higher boost pressures on turbo cars.

More air in and out means a bigger bang.

s/better/louder.

I definitely don't consider "louder" to be better in this case.

Quite ironically, turbochargers remove a big deal of sound from exhaust, because the energy is extracted back to do useful work. Otherwise it would be emitted as noise pollution, which some sociopaths cherish for some reason. I believe exhaust systems can (and most likely are) acoustically tuned to do similar thing by fluid dynamics too.