Yes, a hiring manager may ask that the candidates perform a writing exercise (i.e., responding to a pre-determined question) once the candidates have been found to be the best qualified and are at the end of the hiring process (i.e., at the final interview).
However, a writing sample like that is not scored; it's merely used to inform the hiring manager of the candidate's writing ability. And just like a "real" assessment or test, there are still many aspects to the process that need to be followed:
You will also need to consider:
It is up to your discretion how you (or the hiring manager) collect the writing sample. For example, you could ask the candidate to respond to a question or have them fix a grammatically-incorrect paper. Just keep in mind what writing skill you want to measure in order to guide the development of the assessment.
As with all testing practices, it's paramount to standardize the process, meaning that all candidates who are asked to produce a writing sample are treated the same, given the same question, and so forth.
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