The newel post is the anchor of the handrail on a staircase. If the newel post is loose, the railing and the other posts will be wobbly and unstable. To fix a wobbly newel post you need to bolt it back into the frame of the staircase. Since the newel post is very important to the safety of the stairs, if one bolt doesn't fix the loose newel post you may need to install a second bolt.
Drill a 1-inch-deep hole into the side base of the newel post with the 1-inch-diameter spade. Do not drill straight into the base, but instead angle the drill bit toward the first step so that you can reach the stair frame with the bolt. Stop drilling when the hole is 1 inch deep.
Use the access hole you've made to drill a 7/32-inch pilot hole through the rest of the newel post and into the stair frame. Follow by drilling the newel post portion of the 7/32-inch hole with the 5/16-inch bit to make it wider, but stop widening the hole short of the stair frame.
Screw a 5/16-inch lag bolt that is longer than the hole through the newel post into the hole you've drilled, using a screwdriver or wrench to screw it into place and secure the newel post to the stair frame.
Cut a wood plug for the 1-inch hole by sawing off the end of a 1-inch-diameter dowel rod. Insert the wood plug into the hole.
Cover the plug with wood putty, using the putty knife to smooth it over the surface. Allow it to dry fully.
Sand the wood putty so that you have a smooth surface. Wipe away any dust particles and then paint the putty to match the stairway's finish.