Listen AoD.
I know it sucks when you don't get what you want, but there are more reasonable ways then just nagging.
You have tried the Argument that you need it for school, if I were you I'd keep that in the back of your head.
Now what you gotta do is convince your folks at a very good way.
Try to find the best Price/quality equation, in other words try to find what you need as cheap as possible but still the best possible quality for that price range.
Try to find several options with differing prices, if you can't get that 2,2 GHz processor for a decent price then take it lower to 1,3 GHz just to name an example.
It will show your folks that you are doing research and that you are as economic as possible, also try to talk with your folks at a very easy pace, if you come barging in they will be "scared" by you and turn your request down no matter what the outcome, also (I don't know your exact age so I'll go out from the fact that you are 17) try to find a job and save as much as possible so you can always say that you are willing to pay a part of it, I think your folks will really like that.
If this doesn't help you then I don't know what.
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Oh, dear, How sad, Never mind!!-Battery Sergeant Major Williams
Plus the newest addition!!-