a couple of days ago my friend removed the hose from my airbox now my integra sounds pretty beefy, but i was wondering if this messes up my mpg.

 

You mean from the resonator to the airbox or from the airbox to the throttle body? If you're running without any air filter then your mpgs may soon be the least of your worries.

 

is this "hose" a big beefy thing that looks like a cancerous "U"

 

if it is the resonator, it wont reduce mpg or anything...just is louder and will probably result in you stomping the pedal on the far right more which indirectly may result in loss mpg. if u just removed the intake hose then u are dumb and your motor will be very sad.

 

Letting people do things to your car w/o you understanding or knowing what they did is not a good idea.

 

You're car just lost hp from removing that.