We derive explicit expressions for the correlation coefficients between
$\bar{X}$ and $S^2$ and $\bar{X}$ and
$\frac{n}{(n-1)(n-2)}\sum_{i=1}^n(X_i-\bar{X})^3$ in terms of sample moments.
Using these we show that two tests for normality, proposed by Lin and Mudholkar
(1980) and Mudholkar et al. (2002), can be simplified by using moment
estimators; particularly the sample skewness and kurtosis; rather than the
jackknife estimators previously used. In an extensive simulation power study
the tests exhibit higher power than some common tests for normality against a
wide range of distributions.