Backup contents not showing during Time Machine system restore
I just installed a new hard drive on my Macbook Pro so I'm trying to restore a backup of my old hard drive that was saved on a USB-connected external drive (model: LaCie Minimus). However, when I try to transfer the information during system setup, my backup is not showing as part of the external drive's contents. Only the backup information of two other computers are showing up, but not mine (my computer is called 'Hannah Mercader's Macbook Pro'):
However, if I plug the external drive into another laptop to view its contents, my computer's backup information is clearly in the external drive:
I am positive that I have been backing up my laptop properly using Time Machine. However, the last time that I did try backing up my laptop, it didn't work because my hard drive had failed in the middle of the process (hence the reason for needing to install a new hard drive). I've already tried to Repair the external drive using Disk Utility... everything seems OK but my backups still aren't showing when I try to set up my laptop.
Please help me to restore my last proper backup's contents!
MacBook Pro (15-inch Mid 2009), OS X Mavericks (10.9)