They say it's a folk tune from the Appalachians but provide absolutely no backing for their guess.
They don't say it's from the Appalachians, but that it is 'typical of'. A totally different meaning.

Notes

NEW BRITAIN (also known as AMAZING GRACE) was originally a folk tune, probably sung slowly with grace notes and melodic embellishments. Typical of the Appalachian tunes from the southern United States, NEW BRITAIN is pentatonic with melodic figures that outline triads. It was first published as a hymn tune in shape notes in Columbian Harmony (1829) to the text "Arise, my soul, my joyful pow'rs." It was first set to "Amazing Grace" in William Walker's (PHH 44) Southern Harmony (1835) (see… Read More