Your information is accurate, except your ancestors were imported from Scotland to pacify the Catholic insurrection on the emerald isle. Scot Irish were highly favored immigrants to America, because they were known to be rugged, and facile with the gun, and not too interested in abstractions, and thus were deemed ideal as frontiersmen to pacify the Indians just as they had pacified the Catholic Irish. And that proved to be the case. President Jackson and George Wallace are two prominant Scot Irish American politicians. Perhaps 40 million Americans have some Scot Irish blood as a wild guess. Of course, the clan has been rather totally assimilated at this point. Oh the horror.
Hilbillies are Scot Irish - you can hear it in the music.