so you're holding down the 'F' on the E string and using 1 finger per fret, with your pinkie grabbing that A flat and your ring finger grabbing the G ?
I can think of 2 main reasons you're getting buzzing from the E (assuming your bass is set up ok)...
1. your pinkie isn't stretched out far enough to hold down the string in the optimum position i.e. right behind the fret (which isn't anything to be ashamed of.. it's a long stretch for those of us with normal sized hands).. for the stretch to the G, remember your ring finger is naturally the weaker and less mobile of all your fingers, and it'll always probably always feel a bit of a poor relation to the others

(something about nerve pathways & tendons i believe)
2. maybe it IS stretched out and in the correct place but the amount of finger strength you can apply isn't quite enough to hold it down firmly and let the note ring properly
either way, plenty of practice with 1-finger-per-fret exercises up and down the board will soon cure both of those things... it sounds like a case of a bit more stretch & a bit more strength needed... you'll be surprised how quickly it can be overcome