  We show that the One-way ANOVA and Tukey-Kramer (TK) tests agree on any
sample with two groups. This result is based on a simple identity connecting
the Fisher-Snedecor and studentized probabilistic distributions and is proven
without any additional assumptions; in particular, the standard ANOVA
assumptions (independence, normality, and homoscedasticity (INAH)) are not
needed. In contrast, it is known that for a sample with k > 2 groups of
observations, even under the INAH assumptions, with the same significance level
$\alpha$, the above two tests may give opposite results: (i) ANOVA rejects its
null hypothesis $H_0^{A}: \mu_1 = \ldots = \mu_k$, while the TK one,
$H_0^{TK}(i,j): \mu_i = \mu_j$, is not rejected for any pair $i, j \in \{1,
\ldots, k\}$; (ii) the TK test rejects $H_0^{TK}(i,j)$ for a pair $(i, j)$
(with $i \neq j$) while ANOVA does not reject $H_0^{A}$. We construct two large
infinite pseudo-random families of samples of both types satisfying INAH: in
case (i) for any $k \geq 3$ and in case (ii) for some larger $k$. Furthermore,
in case (ii) ANOVA, being restricted to the pair of groups $(i,j)$, may reject
equality $\mu_i = \mu_j$ with the same $\alpha$. This is an obvious
contradiction, since $\mu_1 = \ldots = \mu_k$ implies $\mu_i = \mu_j$ for all
$i, j \in \{1, \ldots, k\}.$ Similar contradictory examples are constructed for
the Multivariable Linear Regression (MLR). However, for these constructions it
seems difficult to verify the Gauss-Markov assumptions, which are standardly
required for MLR.
