The Hammers secured the short-term signature of Carroll from Liverpool on Friday, and the England international was instrumental in his new side's 3-0 win over Fulham the next day.
The 23-year-old didn't score in that game, but Allardyce believes that the number of goals Carroll does score this season will have a direct correlation to their position come May.
"If Andy scores between ten and 15 goals, it is going to give us a great chance to be out of the relegation zone. If Andy gets 15 to 20 goals, we can get in the top ten because all we have to do there is to produce enough clean sheets to make sure we get points on the board with the goals that he scores," Allardyce told the club's official website.
"You can look at what happened at Newcastle in the first year when Kevin Nolan could feed off Andy. By the time Andy had been transferred to Liverpool, he had scored eleven goals and by the time Kevin had finished the season, he had scored 13 and they were really comfortable in the division."
Carroll has had to pull out of England's World Cup qualifiers due to an injury suffered in the match against Fulham.