2003 Elantra Thermostat Replacement

I have an '03 Elantra GLS, 2.0l engine, 90,000miles & I suspect it needs a new thermostat, its started running very hot yesterday & after allowing it to cool off for almost an hour, I removed the cap & it puked a lot of coolant out.

I had a quick look at where I THINK the 'stat is, but just want to make sure it is indeed in the small housing on the right of the engine (when viewed from the front) where the top hose goes into the cylinder head.

Thanks.