It is not easy to come up with a simple answer to 'prove' that CNC is faster. I did the transition from manual machines to CNC in my business starting just over ten years ago and I know that CNC is much more productive per employee, but I don't have the data to directly compare the two. You can read these threads where I describe some of the advantages of CNC. Definitely a big advantage is that you can machine shapes that are difficult if not impossible to do on manual machines as Algirdas mentions.
But it is not just switching to CNC that provides the best improvement in productivity it is also designing fixtures, programs and work procedure to utilize the CNC machine effectively. This aspect is mentioned in my threads.
http://www.cnczone.com/forums/haas_m...y_my_shop.html
http://www.cnczone.com/forums/haas_m...fixturing.html
http://www.cnczone.com/forums/work_f...fixturing.html
If you have any specific questions I may be able to answer them.
An open mind is a virtue...so long as all the common sense has not leaked out.