  We improve both upper and lower bounds for the distribution-free testing of
monotone conjunctions.
  Given oracle access to an unknown Boolean function $f:\{0,1\}^n \rightarrow
\{0,1\}$ and sampling oracle access to an unknown distribution $\mathcal{D}$
over $\{0,1\}^n$, we present an $\tilde{O}(n^{1/3}/\epsilon^5)$-query algorithm
that tests whether $f$ is a monotone conjunction versus $\epsilon$-far from any
monotone conjunction with respect to $\mathcal{D}$. This improves the previous
best upper bound of $\tilde{O}(n^{1/2}/\epsilon)$ by Dolev and Ron when
$1/\epsilon$ is small compared to $n$.
  For some constant $\epsilon_0>0$, we also prove a lower bound of
$\tilde{\Omega}(n^{1/3})$ for the query complexity, improving the previous best
lower bound of $\tilde{\Omega}(n^{1/5})$ by Glasner and Servedio.
  Our upper and lower bounds are tight, up to a poly-logarithmic factor, when
the distance parameter $\epsilon$ is a constant. Furthermore, the same upper
and lower bounds can be extended to the distribution-free testing of general
conjunctions, and the lower bound can be extended to that of decision lists and
linear threshold functions.
