I appreciate everyone's very helpful responses.
I was invited to an event last night, which I took part in. It was an informal event, but it met the number of sisters in attendance requirement. We went to an Italian restaurant for a sister's birthday, and the girl who invited me to the event at the end of the week invited me on a spur. I was nervous, but I put that aside and sucked it up and went.
I tried to mingle with all of the girls around me, asking them questions about where they were from and assorted things about themselves, as well as giving them a bit of information about myself. I tried not to fall too silent, but there were occasions that I would, so I would push through them with another question.
They invited me back to the house and gave me a tour. While I don't feel as though I talked a lot, there were still quite a few girls I felt comfortable around, and as weird as it sounds, it felt natural being in the house, laughing with those girls. Even if I am not given a bid, I think I have found a few new girls who might become good friends.
When the girl who invited me took me back home, she told me that I had met a lot of the people she had wanted me to meet (such as the president, who was extremely nice), and she believed I had a decent chance at being selected, although she said she could not, of course, tell me that on the record or as a fact. She stated that she felt I had fit in, and she would love to have me a sister.
Like I said, a big thank you to those who helped me out with their responses. Even if I don't become a sister in this house, these are some really great girls, and just because I don't become a sister doesn't mean they cannot be friends. Wish me luck!