No you don't, to be safe, mount psu in case, plug psu into the wall, keep psu off, periodically touch your case as you instal your components. I've done it in shag carpet with socks on without issues. People that say otherwise probably wear tin foil hats
I only bought an antistatic mat when I started my business- I don't think it has changed a single thing for me other than being able to lay motherboards straight onto my table without worrying as much- even though I still don't prefer to do that as much as I prefer just putting it on the box for the mobo.
I've got an IP35 and Q6600 here, if I remember tomorrow, I'll rub the bare CPU on the carpet, on my clothes, and the motherboard on the carpet, then boot the pc
I mean, I would use one. But that's just me. I know what static can do, and I prefer wear anti esd stuff. But you could probably get away without one as long as you're smart and careful.