A VERY LOW TIME C172 PLT DEPARTED THE ARB ARPT TFC AREA IMPROPERLY; THEN FAILED TO EXERCISE PROPER SEE AND AVOID TACTICS; WHICH CAUSED AN NMAC.

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I WAS TAKING MY FATHER FOR HIS FIRST RIDE SINCE BECOMING A PLT. ON MY FIRST DEP I BECAME CONCERNED THAT THE ENG WAS NOT PRODUCING FULL PWR; SO I RETURNED FOR LNDG. AFTER A FULL PWR RUN-UP TO CLR FOULED PLUGS AND 1 TRIP AROUND THE PATTERN TO CONFIRM THE SOLUTION; I DEPARTED NBOUND FROM RWY 6 FOR SIGHTSEEING. SHORTLY AFTER DEP; I HEARD ANOTHER PLANE CALL THE TWR INBOUND FOR LNDG FROM THE N. I CONTINUED NBOUND; LEVELED OFF AT 3000 FT MSL; LOOKING FOR TFC. THE TWR CALLED MY TFC 12 O'CLOCK POS; 3 MI; 2600 FT; BUT I COULDN'T SEE HIM. I TOLD THE TWR I WOULD ALTER COURSE TO THE NW AND TURNED IN THAT DIRECTION. THE TWR THEN CALLED MY TFC AT 12 O'CLOCK; 1 MI; 2600 FT. I STILL DID NOT SEE THE OTHER ACFT. SHORTLY THEREAFTER; THE OTHER PLT; WHO HAD UNTIL THEN ALSO RPTED NEGATIVE CONTACT WITH ME; RPTED THAT HE HAD PASSED UNDER ME WITH 200 FT (OR MAYBE 100 FT; I'M NOT SURE I REMEMBER CORRECTLY) SEPARATION. THE TWR DECLARED MY TFC NO FACTOR AND I CONTINUED WITH MY FLT. I MADE A NUMBER OF ERRORS. I WAS DISTR BY MY FIRST FLT WITH MY FATHER AND BY THE SPARKPLUG FOULING I HAD JUST DEALT WITH. ALTHOUGH I ALTERED COURSE; I WAS TOO FIXED ON MY NORTHERLY DIRECTION TO ALTER COURSE SUFFICIENTLY OR TO INITIATE A CLB. I ASSUMED THE OTHER PLT WOULD PROCEED DUE S AND DSND TO THE PATTERN; WHEN I SHOULD HAVE KNOWN HE WOULD HEAD SW TO JOIN THE PATTERN FOR RWY 6 AND DELAY DSCNT WHILE OVER THE CITY. WHEN I DIDN'T SEE HIM AT 3 MI I SHOULD HAVE CLBED AND TURNED R.