CORP LTT FLC MISHEARS HDG ASSIGNMENT; TURNS INTO FLT PATH OF TURBO ACFT CAUSING TCASII TO ACTIVATE.

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WE WERE BEING VECTORED FOR AN APCH TO ILS RWY 27R AT MIA. ON A HDG OF 090 DEGS; WE WERE GIVEN A HDG CHANGE TO 070 DEGS. THE PF UNDERSTOOD A HDG OF 170 DEGS AND STARTED HIS TURN. I THOUGHT THE APCH CTLR HAD SAID 070 DEGS BUT BECAUSE OF RADIO CONGESTION MY INQUIRY AS TO VERIFICATION OF THE HDG MUST HAVE BEEN STEPPED ON. IN OUR TURN TOWARD 170 DEGS; I NOTICED AN AIRLINE TURBOPROP AT OUR 12 O'CLOCK POS AT 2 MI. I POINTED TFC OUT TO THE PF AND HE IMMEDIATELY STARTED A TURN AWAY FROM TFC AND BACK TOWARD OUR 090 DEG HDG. AS WE WERE MAKING OUR TURN BACK; THE CTLR ASKED IF WE WERE TURNING TO THE HDG 070 DEGS. I STATED THAT WE WERE MAKING A TURN BACK TO HDG 070 DEGS. ABOUT THIS TIME; THE TURBOPROP IN QUESTION HAD A TCASII WARNING AND HAD STARTED A CLB. THE CTLR VERIFIED WITH US THAT WE HAD THE TURBOPROP IN SIGHT. THEN HE ADVISED THE TURBOPROP THAT WE HAD HIM IN SIGHT AND WERE TURNING AWAY FROM HIM. AS CLOSE AS I CAN TELL WE CAME WITHIN A MI OF THE TURBOPROP. A MISUNDERSTANDING OF COM PLUS THE RADIO CONGESTION; ALL CONTRIBUTED TO THIS SIT. I BELIEVE THE PF MADE THE RIGHT DECISION IN TURNING AWAY FROM OTHER ACFT TO AVOID CONFLICT OR COMING ANY CLOSER TO THE OTHER ACFT. I'M SURE WE BOTH WILL GIVE MORE ATTN TO VERIFICATION OF HDG CHANGE.