The E spec is a test method, it tells you how to do it but has no acceptance criteria.
And A product spec will require testing to a method and it will give you the standard acceptance requirements.
In this case since A388 is labeled a practice my guess is that you will need the product spec, practice, and test method in order to have all of the details.
If you are buying something to an ASTM spec and you don't own all of the relevant specs then you have decided that it does not matter. As purchaser it is up to you to enforce the correct testing and certification. If you don't know then obviously you don't care.
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P.E. Metallurgy, Plymouth Tube