B717-200 CREW TOOK A XMISSION AND CLRNC FOR ANOTHER ACFT IN SDF CLASS C.

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I WAS CAPT OF ACR X FLT; ABCD DEPARTING SDF FOR STL. THIS WAS THE FIRST FLT OF THE 3RD DAY OF A 4-DAY TRIP. WE HAD HAD A VERY SHORT LAYOVER THE NIGHT BEFORE AND A VERY EARLY WAKE-UP THIS MORNING. I GOT MAYBE 5 HRS OF SLEEP. WE HAD AN FAA AIR SAFETY INSPECTOR RIDING IN THE JUMP SEAT. SHE WAS AN AVIONICS TECHNICAL TYPE WHOSE NORMAL DUTY WAS OVERSEEING THE CARGO MAINT FUNCTIONS AT SDF. SHE WAS NOT A PLT. ALTHOUGH PERSONALLY PLEASANT; SHE HAD ALREADY IDENTED SOME CABIN MAINT DISCREPANCIES WE HAD TO DEAL WITH BEFORE DEP. SHE ALSO WAS OBVIOUSLY A VETERAN AIRLINE OBSERVER AND WELL AWARE OF WHAT WAS GOING ON. I WAS THE PF. WE DEPARTED RWY 35R. THE PRE-TKOF CLRNC WAS RWY HEADING TO 5000 FT; EXPECT RADAR VECTORS ON COURSE. THE ACFT IMMEDIATELY IN FRONT OF US WAS A BIZJET WHOSE CALL SIGN I THINK INCLUDED A 'CD' SOMEWHERE. TFC WAS ALL BUT ZERO. WE WERE DEFINITELY THE ONLY ACR CALL SIGN ON THE FREQ. SHORTLY AFTER TKOF AT ABOUT 3000 FT; I HEARD A RADIO XMISSION THAT WAS '...TURN L HEADING 250 DEGS; CLB AND MAINTAIN 10000 FT.' I WAS MENTALLY BUSY WITH HAND-FLYING THE 3RD-4TH SEGMENT CLB TRANSITION AT THE MOMENT THE CTLR SPOKE AND I MISSED THE CALL SIGN. I DOUBT IT WAS BLOCKED; IT JUST DIDN'T REGISTER WITH ME AND I FIRST 'TUNED IN' AT THE WORDS 'TURN L.' THE FACT I TUNED IN AT ALL INDICATED TO ME THAT I SUBCONSCIOUSLY HEARD MY CALL SIGN AND IT WAS INDEED FOR ME. THE FO READ BACK 'ACR X ABCD; TURN L HEADING 250 DEGS; CLB AND MAINTAIN 10000 FT.' THE VECTOR AND CLB WAS EXACTLY WHAT WE HAD BEEN EXPECTING BASED ON OUR RTE AND OUR MANY TRANSITS OF SDF IN THE PAST. I HAD NO DOUBT THE CLRNC WAS FOR US. I HEARD NO OTHER PARTIAL RADIO XMISSIONS IMMEDIATELY BEFORE OR AFTER OURS. THE CTLR SAID NOTHING AFTER THE READBACK. WE TURNED THE 100 DEGS L AND CONTINUED THE CLB. JUST AFTER GETTING STEADY ON THAT HEADING AND PASSING ABOUT 6500 FT; THE CTLR ASKED 'ACR X ABCD; DID YOU TURN?' THE FO REPLIED 'I READ BACK HEADING 250 DEGS; CLB AND MAINTAIN 10000 FT.' THE CTLR SAID 'THAT WAS FOR ANOTHER ACFT. MAINTAIN 8000 FT FOR TFC AND PRESENT HEADING IS OK.' I LEVELED AT 8000 FT. WE DISCUSSED THE CLRNC AMONG OURSELVES; INCLUDING THE FAA INSPECTOR. BOTH THE FO AND THE FAA INSPECTOR AGREED THEY WERE 100% CERTAIN THE CLRNC WAS FOR US. I WASN'T 100% SURE SINCE I HADN'T REALLY HEARD THE CALL SIGN; BUT I TEND TO BELIEVE IT WAS FOR US. THE CTLR EVENTUALLY CONTINUED OUR CLB AND HANDED US OFF TO ARTCC AND NOTHING MORE WAS SAID. PRESUMABLY OUR ACTIONS DIDN'T RESULT IN A TFC CONFLICT. WE CERTAINLY NEVER HAD A TCASII ALERT; NOR DID WE SEE ANY RELEVANT TARGETS ON TCASII OR VISUALLY. LESSONS LEARNED: 1) WHEN YOU'RE NOT 100% SURE IT'S FOR YOU; ASK. 2) IT'S ALSO POSSIBLE THE CTLR GOOFED AND SAID THE WRONG CALL SIGN; OR COMPLETELY FORGOT THAT HE GAVE US THE CLRNC AND WAS SURPRISED AT OUR TURN. 3) AN FAA INSPECTOR; NO MATTER HOW NICE; IS A MAJOR DISTRACTER IN THE COCKPIT. WHEN AN FAA INSPECTOR IS ABOARD; IT'S A PAIN AND A SOURCE OF DISTR; AND WHEN THEY'RE NOT; WHETHER ONE IS ABOARD ONCE OR TWICE A YR OR NOT MAKES NO DIFFERENCE IN HOW I FLY THE REST OF THE TIME.