There are some ways to install and apply a style-file in a mac.
 Here 
http://slavecache.blogspot.com/2010/09/installing-sty-files-for-latexlyx.html
"for the right directory to drop your files in. Under Mac the appropriate directory is
    /usr/local/texlive/2009/texmf/tex/latex/my_new_styles
Now run sudo texhash (or sudo mktexlsr or texconfig[-sys] rehash) and check with kpsewhich your_style_file if it worked.
LyX should be able to find your sty-files now. " 
This website describes how to do it by using absolute path.  But for a mac beginner, he will be very hard to find that path.  We shall change to some other ways.
The following is what I usually use:
1. go to Macintosh HD/Library/Tex  (very easy to do this step)
2. click "root" to enter 
3. click "texmf" to enter
4. click "tex" to enter
5. click "latex" to enter
But this is a root directory, which means you cannot make a sub-directory here
Thus we need to find other way out.
6. Go to any other folder rather than here,
7. make a new folder called "my_new_styles"
8. copy your style-file (*.sty) into this folder
9. copy this folder to the place where in step 5 you goes
10.  do "sudo mktexlsr " in terminal
11. reconfig Lyx and restart
Now you can use your new style-file.
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