I found the following family on the 1851 England census in Hetton le Hole, Durham:
John Bates, head, 31, pitman, born Dubmoor, Durham
Hannah Bates, wife, 26, born Benton, Northumberland
William Bates, son, 6, born Benton, Northumberland
Thomas Bates, son, 4, born Benton, Northumberland
Joseph Bates, son, 2, born Castle Eden, Durham
Jane Ann Bates, daughter, 11 months, born Benton, Northumberland
Mary Pringle, visitor, 8, born Benton, Northumberland
Piece: 2393; Folio: 373; Page: 12
I know his occupation is different but it's a possibility. I couldn't find the family after 1851.
Jacquie