A B757 FLT CREW EXPERIENCED A TCASII TA WHILE DSNDING INTO JAC.

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APPROX 50 MI S OF JAC; ZLC INFORMED US THAT ANOTHER ACFT WAS SHOOTING A PRACTICE APCH TO RWY 18 AT JAC AND THAT WE SHOULD SLOW TO AVOID HOLDING. WE SLOWED AND CHANGED OUR REQUEST FROM VOR-DME RWY 36 WITH VISUAL TO RWY 18 TO THE ILS RWY 18. CLRED DIRECT DUNOIR. JUST PRIOR TO DUNOIR WE ASKED IF WE WERE TO HOLD. WE WERE THEN GIVEN HOLDING INSTRUCTIONS TO HOLD AT QUIRT 14000 FT EFC OF XA15. TIME NOW XA05. THIS OTHER LIGHT TWIN CANCELS IFR. CTR CLRS HIM TO ENRTE FREQ AND CLRS US FOR THE ILS RWY 18. WE ARE ABOUT 3 MI SHORT OF QUIRT AT THIS TIME; WE DSND AND JOIN THE LOC AT QUIRT. WE DO NOT SEE THE OTHER ACFT AT THIS TIME. ZLC SWITCHES US TO JAC TWR. WE ASK THE TWR WHERE THIS OTHER ACFT IS. THEY DO NOT KNOW AND HAVE NOT TALKED TO HIM. THEY TRY TO REACH HIM ON TWR FREQ. NO HELP. THEY CALLED CTR TO SEE IF HE IS STILL UP THEIR FREQ. NO HELP. AT ABOUT 10 MI ON FINAL WE RECEIVE A TCASII ALERT THAT INDICATES AN ACFT AT .01 NM BEHIND US AND 700 FT BELOW US. WE RPTED A MISSED; CONTACT TO ZLC AND WITH TWR. ENTERED DOWNWIND FOR VISUAL TO RWY 18 AND LAND UNEVENTFULLY. TALKED TO LIGHT ACFT PLT ABOUT CONTACTING TWR AND HE SAID HE DID NOT HAVE TO UNTIL 5 MI; SO HE DIDN'T. ZLC LOSES RADAR ABOUT 13000 FT; SO THEY DID NOT KNOW WHERE HE WAS AT OR WHAT HE WAS GOING TO DO. CONTACTED TWR AND ZLC SUPVR AND EXPLAINED THE PROB.