Assign values of Vertex Normals

Question

I would like to change all the normals of selected vertices in my scene to be (0,0,1). I am attempting to do this on a single mesh as a test scenario using the following code:

import bpy

mesh = bpy.context.object.data

selected_verts = [v for v in mesh.vertices if v.select]

for v in selected_verts:    
    print(f"Before Vertex Normal: {v.normal}")
    v.normal = [0,0,1]
    print(f"After Vertex Normal: {v.normal}")

When doing this, the value appears to change but the normal in the scene remains unchanged. Using Blender 2.83LTS

EDIT - subbed out the averaging routine for a Vector(0,0,1) - this works for my needs since all my Vectors are either UP or DOWN -

#Evaluate face Normals - if negative - flip
for obj in selected:
    bpy.ops.object.select_all(action='DESELECT')
    obj.select_set(True)
    bpy.context.view_layer.objects.active = obj
    bpy.ops.object.editmode_toggle()

    bm = bmesh.from_edit_mesh( bpy.context.object.data )

    # Reference selected face indices
    bm.faces.ensure_lookup_table()
    selFaces = [ f.index for f in bm.faces if f.select ]

    # Positive vector
    up = Vector([0,0,1])

    # Calculate the dot products between the Up vector and each face normal
    dots = [ up.dot( bm.faces[i].normal ) for i in selFaces ]

    # Reversed faces have a negative dot product value
    reversedFaces = [ i for i, dot in zip( selFaces, dots ) if dot < 0 ]

    # Deselect all faces and (later) only select flipped faces as indication of change
    for f in bm.faces: f.select = False
    bm.select_flush( False )

    for i in reversedFaces:
        bm.faces[i].select = True
        bm.faces[i].normal_flip()  # Flip normal

    bm.select_flush( True )

    bpy.ops.object.editmode_toggle()

bpy.ops.object.select_all(action='DESELECT')