Hi Hedy,
I have to agree with Casorati in that it would be much better if you had already taken real analysis (or at least some other heavily proof-oriented course). I am assuming that the courses on mathematics engineering you took are mostly based on Calculus, differential equations, complex analysis, am I right? I believe a course in Mathematical Statistics or theoretical probability would be a big plus too.
One option is spending more time on your current program and take more pure math courses. Another is looking for programs in between business and statistics, since you have done a lot of coursework in there (and probably have recommendation letters from people there). I am talking about Operations Research, Decision Sciences, or some more quantitative programs of Business/Marketing and such. You might end up doing a lot of statistical research in a Marketing dept, for instance, although you will be in a very different environment.
There's probably some programs in statistics that won't require from you a very strong background in math, but I don't know much about them. Also, maybe your recommendation letters will be able to convince the admissions that you are able to grasp math without problem, although it might be hard for the admissions to believe that you know what deep math if you haven't been exposed to it via coursework or research.
Sometimes people don't understand when I talk about deeper math and think it is just undergrad math (Calculus and so on). To make sure what I mean by it, just take the book of real analysis by Rudin and read the first and second chapters. If you don't know the stuff in it, it's ok. But if it scares you, then it is a sign that you need to learn it.
I don't want you to loose hope on your application. Reality depends a lot on small details that I and grad Cafe might not address properly. So I recommend you to talk to as many people as you can (students and profs in STAT depts you would like to go to), preferably people that know you.