Distinguishing a biased coin with a small set of tests
Yes. Such families are called "averaging samplers", and there are plenty of constructions for them. You can find a more information about them (and about the more general notion of sampler) in this survey.

The notion of averaging samplers was introduced in this paper, which also showed that they are equivalent to randomness extractors. There are many constructions of extractors, and you can use any of them to construct samplers.

Here are a few relatively simple ways to construct such families:


Take a random family, as pointed out by Emil Jerabek in the comments.
Construct a family using a $k$-wise independent hash function, as pointed out in the comments by D.W.
Construct a family by considering an expander graph over the vertex set [n] and taking the neighborhood of radius $r$ around every vertex (for some sufficiently large values of $r$) - see for example this lecture. Alternatively, take the family obtained from the set of random walks of length $r$ on the graph. -- see for example this lecture  .