Usually 1" shafts take either 3/16" or 1/4", 5/16" sounds way too big.
Previous advice on this same recurring question suggests that either you measure your stuffing box (back it off, center it and measure the distance between the shaft and the box), or buy both 3/16" and 1/4", try them and either return the unused (sometimes difficult if you buy a cut length like Ultra X) or give it to a friendly skipper in your marina.
Good luck.