You Do Need power plug adapter while traveling from Cayman Islands to Samoa.

Here is Why You Need An Adapter

The outlet and plug type used in Samoa is I and in Cayman Islands are A and B. This is how they look like.

Cayman Islands

In Cayman Islands, outlets of type A and B are in use.

Type A

Type B

Samoa

In Samoa, outlets of type I are in use.

Type I