Is the Sigma EX 70-200mm f/2.8 APO HSM DG OS fully compatible with the nikon D3000?

Does the auto focus work and also is there's a similar cheaper version?

If you go to the Sigma website, you will see that the HSM designation tells you that yes, indeed the lens has and auto-focus motor in the lens, so it will work on your D3000

It is on sale in the States for under $1, 250 or about $1, 150 less than the Nikkor AF-S 70-200 mm f/2.8 lens

Yes, as long as you buy the Nikon mount version of the lens. HSM = high speed motor, so it will autofocus on your camera.

This is a great sports lens, and it is the cheaper version (the Nikon 70-200mm f/2.8 is $2, 400).

You can get a Tamron version for around $800~900 I think, but I would stay with Nikon or Sigma as the Sigma lens is known to be one of the better ones they make.

The only other alternative is the Nikon AF 80-200mm f/2.8 at $1, 200, which is the lens I have (also a great lens), but it won't autofocus on your camera (only on D90, D7000, and above).

The lens is compatible with any Nikon, it is your camera that is not compatible with half of what Nikon makes.

If you had actually read any specs on the lens, not just the price tag, you'd already know this.

Yes the Sigma EX 70-200mm f2.6 APO HSM DG OS will be fully compatible with your D3000. Enjoy!