I purchase a brand new home and I immediately noticed there was a loud noise inside the wall when the furnace fan kicks in every time (every 5 to 7) minutes, all night long and all winter long, is loud enough that it wakes me up all the time, the home inspector that came to do an inspection thinks is the duct work that is too tight. If this is correct, can this be easily fixed, or fixed at all, and how? How much would this cost? Which trade should be do the job?

If it is a brand new home it is under warranty have the builder fix it. OR contact a local HVAC mechanical contractor and get ON SITE advice since we cannot hear or see the entire situation from here. Sheet metal has been known when improperly installed to make an expansion and contraction noises like banging or popping. This usually is not an easy or cheap fix since your problem will possibly involve the following trades. HVAC mechanical, drywall installer, painter (good luck getting a paint match without painting the entire room), possible carpentry and electrical might have to be moved. Permits or building inspections might also have to be obtained depending upon your jurisdictions laws/rules and regulations along with the fact that it is remotely possible that structural design might need to be changed.

If all else fails or you want to be lazy get a lawyer and sue the builder if he will not fix it because this might not be cheap. (Labor/material prices vary across the nation and there is no way to do an estimate here without actually visiting ON SITE to check the problem.

Good luck with your situation.

PS: It has also been known that a properly placed sheet metal screw could stop the expansion and contraction but you would need to get lucky and find a older experienced HVAC service/installer to accomplish that bit of magic. I used to track down noises like yours with a special listening device that had head phones. After finding the location of the loudest noise I would sink a 3 inch sheet metal screw right through the drywall and counter sink it allowing for a very small amount of drywall compound to cover the repair (kind of like repairing a nail pop). Usually a finger wipe of the compound did the job of coverage. This was a very cheap way to do the repair but costly for my time.