Final answer:
The American Red Cross is not a public agency; it is a nonprofit organization, unlike the other options which are government-related bodies.
Step-by-step explanation:
The agency that is not a public agency is the American Red Cross. The American Red Cross is a humanitarian organization that provides emergency assistance, disaster relief, and education in the United States; it is a nonprofit organization, not a government agency. In contrast, the National Institute of Nursing Research is part of the National Institutes of Health, the Centers for Disease Control and Prevention is a federal agency within the Department of Health and Human Services, and the State board of nursing is a state governmental body tasked with regulating nursing practice.