The wheel bearings on a 2003 Toyota Corolla are part of an assembly that includes the hub. The axle carrier contains this assembly within it, as well as the rear brake drum assembly on each wheel. You need to replace the hub and bearing assembly as a unit if the hub shows any signs of excessive run out or is loose. Inspect the hub and bearing assembly on both rear wheels to see if you need to replace both of them.
Raise the Corolla's rear end and support it on jack stands. Remove the wheel that corresponds with the bearing and pull the brake drum off the brake assembly.
Remove the four bolts connecting the hub and wheel bearing to the axle beam with a wrench; access these bolts from the rear side of the brake assembly's backing plate.
Remove the hub and bearing assembly out of its seat within the axle carrier, and out through the drum brake assembly.
Place the replacement wheel bearing and hub assembly on the axle carrier within its seat, and then align it so the mounting holes line up with the holes in the brake drum assembly.
Apply the bolts and tighten them to 45 foot-pounds.
Slip the brake drum back on to the axle and brake assembly.
Reconnect the wheel and lower the car off the jack stands.