Essex is part of East Anglia.

No

"East Anglia initially consisted of Norfolk and Suffolk, but upon the marriage of the East Anglian princess Etheldreda, the Isle of Ely also became part of the kingdom. Formerly a standard statistical region, for administrative purposes East Anglia now forms part of the East of England government office region and is defined as a Level 2 Nomenclature of Territorial Units for Statistics, comprising the counties of Norfolk, Suffolk and Cambridgeshire including the city of Peterborough unitary authority area."

That is East Anglia, you are confusing it with the larger East of England government office region and I quote again

"Before the creation of the East of England region there had been a smaller standard statistical region of East Anglia which did not include Essex, Hertfordshire or Bedfordshire. They had previously been included in the South East region. The East of England civil defence region shared the boundaries of the current region."