Given the information about toppling, there has to be a relationship between the co-efficient, the center-of-mass and the force.
If the chair topples, then this means that the sum of all forces is positive in a given direction with respect to the center of mass and the moment of inertia. When forces do not cancel each other and a net force exists in some direction, (and provided there is no other stress involved), then the object should move in that direction.
To understand your question in a deep way, you should look at how the center-of-mass and the moment of inertia help explain the toppling over in terms of all forces including the frictional force. Based on these relationships, a relationship should be able to be derived for the coefficient in order to explain toppling.
Although your moment of inertia will be a lot simpler than many other examples (especially in 3D with complex geometries), this linking of concepts will help you in further physics courses and give further intuition between physical forces and the geometries involved.