With some difficulty I found Timothy and family in 1871 - they are indexed under the surname Toy.
RG 10 / 3014 / 21 / 36
Hill Top, Easington
Timothy Toy, head, mar, 53, ag lab, Ireland
Bridget do, wife, mar, 30, do
Timothy do, son, 14, coal miner, do
John do, son, 12, Durham Spennimoor
William do, son, 10, Yorkshire ?Daw green
Mary do, daur, 6, do do
Margaret do, daur, 4, Durham Spennimoor
John Burke, son in law, 5, do Blaydon
Ann Toy, daur, 9 mo, do Easington
So judging by this I think the family Keyboard found looks like the right one