I discourage you from sending an LOI unless you're already holding an acceptance to a very desirable school. An LOI when you have no other acceptances is meaningless: saying "I'll go to your school over all others if you accept me" doesn't mean anything if you don't have any other options anyway. With very few exceptions, an LOI is only valuable if it's something like "Hi Less Prestigious School, I got accepted to Yale but I really want to stay in my home state of Not Connecticut, so please accept me and I'll dump Yale for you." It's a way of telling school "love me now or lose me forever," and that doesn't work if you have no better alternatives.
As
@Goro said, some schools like to see applicants grovel; a few others, like Mayo, are in barren tundra wastelands and like it when people show specific interest in joining them there. For most of us, though, we assume that you're a) lying and b) desperate unless you give us a compelling reason to believe otherwise. Like, for example, offering to dump Yale for us.
Go to
HomeSkool's Guide to Letters of Intent to see this in graphical form!