Masters in mathematics after bachelors in mechanical engineering
I think your plans for your future are quite reasonable, but my thoughts count for very little. Many engineers have a mathematical bent of mind and I believe many mathematics departments (especially in applied fields like statistics) will recognize that. An example of an engineer who became a mathematician is: Raoul Bott. In all probability, you are not going to be Raoul Bott (prove me wrong), but I think it's worth pursuing the career goals you want. Your job, is to make a convincing case for yourself and to find the right department for you. You can do this by talking with people who have achieved what you are trying to achieve, getting their feedback on their department and your application (especially statement of purpose).