When can an injured player legally be replaced in a basketball game?

A player who is injured can be legally replaced after a time-out is called by their team. This rule ensures that the game can continue smoothly without undue delays caused by injuries. When a time-out is granted, it provides an opportunity for the coach to make changes, including substituting an injured player. This is an essential aspect of game flow and management, allowing teams to strategize and maintain player safety simultaneously.

The other options do not accurately reflect the rules regarding player substitution for injury. A player cannot only be substituted at the end of a quarter, as this would hinder a team's ability to manage injury mid-game. The referee's final decision is indeed important, but it must follow the context of a time-out for a substitution to occur. Reporting an injury may be necessary for certain procedural aspects, but it does not directly dictate when a player can be replaced during the game. Therefore, the answer highlighting time-out as a condition for replacement is aligned with the rules governing player substitutions in basketball.