In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$, using approximations obtained from hypergeometric functions. These results are very close to the best possible using this method. We are able to obtain these results by determining very precise arithmetic information about the denominators of the coefficients of these hypergeometric functions.Improved bounds for the Chebyshev functions in arithmetic progressions $\theta (k,l;x)$ and $\psi (k,l;x)$ for $k=1,3,4,6$ are also presented....