[no chinese keyboard]
one can be interested in statistics by having a couple of applied statistics course (like from business school),
but without some decent mathematical backgroud (namely at least the four course mentioned above), it does mean that one dont have a good-enough preparation for a master in statistics, as doing statistics is not only pluging in the numbers into the program, but also analysis based on data and comparsion across different method, and (where maths comes-in) the proof of different properties of different estimator.
as long as you are going to have them all before graduation, it will be fine anyway. they are most likely listed as a "must" in any statistics master application.
As far as i remember, Yale, chicago also have master in stat, just to name some.
In fact, a applied nature from the stat department don't sounds nice to me: you need an application area.
for someone from that stat/math side, they have good technique, and the major shortcoming will be more likely they dont understand the things they have to analysis (e.g. in finance dont know much about finance, in biomedical they dont know enough about the medi stuff)
for someone from the............um, make an example, business analytics, the graduate are more likely to have better concept on business analysis, but relativly limited stat ablitity.
=> if one want something with applied nature, better try something in other department first........