I just moved in a student flat where several students are using the same WiFi connection. A few days ago the internet was very slow and a guy (son of the owner) came in to ask if I was downloading. He could see I was using 'something that looked like a steaming or downloading site'. It really annoyed me, because I believe I just don't want others checking up on me. I've downloaded Tor, but never really used it. Now I wonder how anonymous I am in the local WiFi spot at home if I use it. I think they can check up on IPs or something, he could see I have a Macbook and I'm the only one with a Macbook in the building. I want to know that if I use a secured connection via Tor other within the network can't see what sites I'm visiting, whatever sites that my be. I don't care if the internet provider knows what I'm doing, but rather not my neighbors since its just none of their business.
If you use Tor or a VPN they cannot detect easily (that is without compromising your computer) which sites you visit, but they can still see how much bandwidth you use. And while they will not see the sites you visit they can just suspect the visiting of special sites from the fact that you try to hide it.