  Generalized t-tests are constructed under weaker than normal conditions. In
the first part of this paper we assume only the symmetry (around zero) of the
error distribution (i). In the second part we assume that the error
distribution is a Gaussian scale mixture (ii). The optimal (smallest) critical
values can be computed from generalizations of Student's cumulative
distribution function (cdf), $t_n(x)$. The cdf's of the generalized $t$-test
statistics are denoted by (i) $t_n^S(x)$ and (ii) $t_n^G(x)$, resp. As the
sample size $n\to \infty$ we get the counterparts of the standard normal cdf
$\Phi(x)$: (i) $\Phi^S(x):=\operatorname {lim}_{n\to \infty}t_n^S(x)$, and (ii)
$\Phi^G(x):=\operatorname {lim}_{n\to \infty}t_n^G(x)$. Explicit formulae are
given for the underlying new cdf's. For example $\Phi^G(x)=\Phi(x)$ iff $|x|\ge
\sqrt{3}$. Thus the classical 95% confidence interval for the unknown expected
value of Gaussian distributions covers the center of symmetry with at least 95%
probability for Gaussian scale mixture distributions. On the other hand, the
90% quantile of $\Phi^G$ is $4\sqrt{3}/5=1.385... >\Phi^{-1}(0.9)=1.282...$.
