I am asked this question all the time, mainly because I have an MBA and I am running a big data project. The answer is, it depends. It depends on the person and what their skill sets are.
Normally the people asking me this question come from IT. They tend to feel that since they run the hardware, they should also be in charge of the data analysis portion of big data. I fully agree that IT should be running the actual physical clusters, and they do where I work. Where we tend to bump heads is when it comes to the data analysis. We analyze all sorts of customer facing data, sales, economics, store traffic and online traffic; most of the questions we are trying to answer are actually customer pain point questions. These are not IT issues, these are business issues.
I also have an analyst background so I know how to analyze data. Also, many MBA programs today have some sort of data modeling class and stats class as part of the core curriculum. A decent MBA should have the basics to know their way around a data analysis system. Of course this could be said about many master level programs. The thing is, who should be doing the data analysis, really all depends on what are the questions you are trying to solve.
If the questions are clearly business and customer focused, an MBA is a strong candidate for that job. If you want to know about ocean currents, probably not. One thing that people need to realize is that the data analysis portion of big data is not really and IT role. The person who fills that role, their skill sets are to be determined by the types of questions you are wanting ask and solve. Until you know the questions, you can’t say who is the best person for the job.