Let Lq,µ, 1 ≤ q < ∞, µ ≥ 0, denote the weighted Lq space with the classical Jacobi weight wµ on the ball B d . We consider the weighted least ℓq approximation problem for a given Lq,µ-Marcinkiewicz-Zygmund family on B d . We obtain the weighted least ℓq approximation errors for the weighted Sobolev space W r q,µ , r > (d + 2µ)/q, which are order optimal. We also discuss the least squares quadrature induced by an L 2,µ -Marcinkiewicz-Zygmund family, and get the quadrature errors for W r 2,µ , r > (d + 2µ)/2, which are also order optimal. Meanwhile, we give the corresponding the weighted least ℓq approximation theorem and the least squares quadrature errors on the sphere.