  For testing independence it is very popular to use either the
$\chi^{2}$-statistic or $G^{2}$-statistics (mutual information). Asymptotically
both are $\chi^{2}$-distributed so an obvious question is which of the two
statistics that has a distribution that is closest to the
$\chi^{2}$-distribution. Surprisingly the distribution of mutual information is
much better approximated by a $\chi^{2}$-distribution than the
$\chi^{2}$-statistic. For technical reasons we shall focus on the simplest case
with one degree of freedom. We introduce the signed log-likelihood and
demonstrate that its distribution function can be related to the distribution
function of a standard Gaussian by inequalities. For the hypergeometric
distribution we formulate a general conjecture about how close the signed
log-likelihood is to a standard Gaussian, and this conjecture gives much more
accurate estimates of the tail probabilities of this type of distribution than
previously published results. The conjecture has been proved numerically in all
cases relevant for testing independence and further evidence of its validity is
given.
