Which policy did Roosevelt implement that emphasized diplomacy backed by military power?

The Big Stick Policy is the correct answer because it encapsulates Theodore Roosevelt's approach to international relations, which emphasized the importance of diplomacy complemented by the potential use of military force. This phrase "speak softly and carry a big stick" reflects Roosevelt's belief that negotiations and peaceful diplomacy were essential, but they should be backed by the readiness and capability of military intervention if necessary.

Roosevelt applied this policy notably in Latin America, where he asserted the United States' influence and intervention rights, particularly through the Roosevelt Corollary to the Monroe Doctrine. This corollary was a specific extension of the Big Stick Policy, but the latter encompasses a broader diplomatic strategy beyond just hemispheric concerns.

Other policies such as Dollar Diplomacy focused primarily on economic investments to secure interests, while Progressive Reform was rooted in domestic change, focusing on social justice and political reform. Therefore, the Big Stick Policy most directly represents the combination of diplomacy reinforced by military strength during Roosevelt's presidency.