To determine the shortest route between two hospitals, which geoprocessing tools collection is most appropriate?

The most appropriate choice for determining the shortest route between two hospitals is the Network Analyst Tools. This collection of tools is specifically designed for solving transportation-related problems, such as finding the shortest path, routing, and service area analysis. When looking for the shortest route, Network Analyst Tools allow users to model the transportation network, taking into account various factors such as road types, distances, and travel times.

In contrast, the other options do not cater specifically to routing analysis. The Proximity toolset focuses on evaluating how features relate to each other in terms of distance, which is not suitable for finding a route. The Statistics toolset is centered around summarizing and analyzing data rather than solving routing problems. Lastly, the Overlay toolset is used for combining multiple layers of spatial data to analyze relationships; it does not provide the necessary capabilities for determining shortest paths or routes between two points.

Therefore, using Network Analyst Tools is essential for accurately and efficiently calculating the shortest route in this scenario.