The R portion of example 1 worked fine, however, when I compiled the .tex file, I received the error below.
!Latex Error: 'Sweave.sty' not found
After spending hours pouring over numerous forums, I found the reason why I received this error and the likely reason that you're receiving this error as well.
My answer below is the initial answer that I was looking for. Feel free to add comments and explanations, however, keep in mind that the instructions are simply to fix this error and allow Miktex to function properly when calling the Sweave package.