But, as shown in Vaughn's photos, you are not talking about a "Morse Taper". The illustration is of the shorter "Jacobs Taper", which is only used for mounting chucks to DP spindles. (Carey's link is about Morse Tapers) Morse tapers are are longer (between 3 & 4 inches long) and used on chuck arbors AND heavy duty drill bits and reamers. The drill press spindle will have to have a MT bore to accomodate the MT arbor. It will also have a slotted hole in the side of the spindle for the drift. Those are made to be readily changed using that tapered drift.
If you would, please Cynthia, show us a picture of your spindle end and your chuck (the end view Vaughn showed). That will help clear up any misinformation on mounting your new chuck. And in any case, Cliff makes a very good point of retracting the jaws all the way in so you are not pounding on them to get the chuck seated.
Best of luck with your new tool, and do be careful.
Aloha, Tony