  We show that the set of numbers that are $Q$-distribution normal but not
simply $Q$-ratio normal has full Hausdorff dimension. It is further shown under
some conditions that countable intersections of sets of this form still have
full Hausdorff dimension even though they are not winning sets (in the sense of
W. Schmidt). As a consequence of this, we construct many explicit examples of
numbers that are simultaneously distribution normal but not simply ratio normal
with respect to certain countable families of basic sequences. Additionally, we
prove that some related sets are either winning sets or sets of the first
category.
