Here is an alternate solution that has the advantage of plug&play instant gratification. Sylvania (brand name) has a LED replacement for our famously anemic #921 back up bulbs that is easily twice as bright as the standard incandescent bulb. It is marketed as a Super Bright LED in the ZEVO line and has a lifetime warranty. It has six LEDs mounted radially, draws 2.5 watts at 12.8 volts. It is no longer (deeper) than a standard #921 bulb and is only a little bit bigger around, so clearance inside the rear light housing is not an issue.
Wal-Mart does not have them in the stores, but offers them in the web site for $22.99 plus $5.12 shipping. O'Reilly Auto Parts has them in the store (instant gratification) for $24.99 (cheaper).
It's a good solution, inexpensive, installation is a snap (use of dielectric grease is recommended), and they're warrantied forever.