We consider the solution of full column-rank least squares problems by means of normal equations that are preconditioned, symmetrically or non symmetrically, with a randomized preconditioner that is computed in lower arithmetic precision. With an effective preconditioner, the solutions from the preconditioned normal equations are almost as accurate as those from the QR-based Matlab backslash (mldivide) command -- even for highly illconditioned matrices. This means the accuracy of the preconditioned normal equations depends on the residual of the original least squares problem, but does not depend on the accuracy of the preconditioner. We present realistic perturbation bounds for the relative error in the computed solutions. This is joint work with James Garrison.