  A Chebyshev-type quadrature for a probability measure sigma is a distribution
which is uniform on n points and has the same first k moments as sigma. We give
an upper bound for the minimal n required to achieve a given degree k, for
sigma supported on an interval. In contrast to previous results of this type,
our bound uses only simple properties of sigma and is applicable in wide
generality. We also obtain a lower bound for the required number of nodes which
only uses estimates on the moments of sigma. Examples illustrating the
sharpness of our bounds are given. As a corollary of our results, we obtain an
apparently new result on the Gaussian quadrature. In addition, we suggest
another approach to bounding the minimal number of nodes required in a
Chebyshev-type quadrature, utilizing a random choice of the nodes, and propose
the challenge of analyzing its performance. A preliminary result in this
direction is proved for the uniform measure on the cube. Finally, we apply our
bounds to the construction of point sets on the sphere and cylinder which form
local approximate Chebyshev-type quadratures. These results were needed
recently in the context of understanding how well can a Poisson process
approximate certain continuous distributions. The paper concludes with a list
of open questions.

Classical Analysis and ODEs;Numerical Analysis;Probability