Shell Scripting, I need to a certain line from a file

I'm not really new to shell scripting, but I've never done anything beyond real basic stuff.  I am trying to write a simple shell script to test a network connection.  It is going to be used by people who don't know anything about computers.  I would like for the user to execute a script that will ping a computer on the network and IF the ping is succesful I would like the script to output "Host Alive", if the ping is not succesful I want it to output "Network Down"

So far I've gotten:

echo "Testing Network..."
ping - c 5 [computer]>results


My thoughts were to output the results to the "results" file and then add an IF statement along the lines of:

IF [Line 6]=*ms
     echo "Host Alive"
else echo "Network Down"

My problem is I don't know how to reference the 6th line of a different file.  

I'm kind of in a hurry to figure this out, I would like to have it working by tomorrow so I'm going to give this question 500 points.  Thank you in advance for any help you can give.  And if you know of a better way of accomplishing the same thing (using a shell script) I'm open to a different way of doing this.  Thanks again.