Can shell-script be setuid ?

["Alex V. Toropov" <alex@ct.spb.ru>]

Can I make a shell script setuid ?
I'v try to do this but it seams that it runs under 
user's id who fired that script (not  under root's as I wish)
When I make a real C program it works as I expect, but 
script doesn't. 
Can anyone explain me where I'm wrong

the script is file starting with
#!/bin/sh

And containing commands like echo with redirection to files in 
"root-only-writable" directory.

TIA Alex