1990 deville exaust
 

I have a question about my exaust on my 90 deville. There is the exaust pipe then there is a rubber hose that comes off the motor and hooks into a piece of pipe that comes off the side of my cat coverter. Where the rubber hose and pipe come together is totally broke on mine the rubber hose is just hanging. Is this hose putting out back pressure my engine needs? is this why my gas mileage is 11 miles to the gallon with a full tune up? and why my idle is rough untill i get up to speed? Information is appreciated. Thanks