I had this problem also, tried installing older versions etc after becoming so frustrated. Searched everywhere on the web for 2 days! This worked for me, couldn't find this fix anywhere:
In my case I didn't have to change any permissions
I changed PHP to 7.0
Rather than uploading the 3 files (prestashop.zip, index.php, Install_PrestaShop.html) to FileZilla, I unpacked the prestashop.zip into a folder and dragged all files over to FileZilla public_html directory. Takes roughly 10 minutes!
Tried my website and it finally got past 67%.
One thing to note when i tried accessing my back office the link wasn't working, link in address bar was ending with "/admin" after installation, just copy the folder name "adminxxxxxxxxx" generated in FTP and paste in address bar.
Seems as if everything's working fine.
I'm hosting with 123Reg Linux Essentials.