  We consider a dataset $S$ held by an agency, and a vector query of interest,
$f(S) \in \mathbb{R}^k$, to be posed by an analyst, which contains the
information required for certain planned statistical inference. The agency
releases the requested vector query with noise that guarantees a given level of
Differential Privacy -- DP$(\varepsilon,\delta)$ -- using the well-known
Gaussian mechanism. The analyst can choose to pose the vector query $f(S)$ or
to adjust it by a suitable transformation that can make the agency's response
more informative. For any given level of privacy DP$(\varepsilon,\delta)$
decided by the agency, we study natural situations where the analyst can
achieve better statistical inference by adjusting the query with a suitable
simple explicit transformation.
