A couple of things come to mind when I see this video:
First, she says that she's not talking about her friends, but about "random people" who she doesn't know. This doesn't make it right for her to talk about Asians or about anyone that way.
Second, yes, she has a right to say what she said. This is America, where people can say what they want, when they want. That's what makes America so beautiful. But it's also what makes America difficult, since people tend to make statements like these and immediately cite freedom of speech to defend such statements. That said, she has every right to have made that video, even if it's not politically correct. It may not be right, but she has her opinions.
Second, I believe there is such a thing as ignorance versus stupidity. Ignorance is not knowing about something; in this context, the girl's rantings might not be terrible because she simply would not have known any better. However, there is a such thing as stupidity when you know about things and still choose to make offensive statements. Having been admitted into a university - UCLA, no less - I would trust this girl has learned the difference between Asian cultures and cultures in general, at least enough to not group Asians together as rude people who say "ching chong". She should also know that what Americans view as rude might be seen as polite in other cultures, and vice versa. So in this context, I would say that she made a statement that was not in her best judgment.
Third, I have to admit that I'm a bit biased in this case because I am an Asian-American too, but I think even as a person, I know more than to make rants like the one this girl made. Yes, it's a bit blown out of proportion, but that doesn't change the fact that there are countless people who think these kinds of things on an everyday basis. So blond, ranting girl aside, it's everyone's job to change the way that people see the world.