Since $y$ is directly proportional to $x^2$, we can write $y = kx^2$. Given $y = 50$ when $x = 5$, we find $k = 2$. Thus, $y = 2x^2$. As $x$ is inversely proportional to $z$, $x = \frac{m}{z}$ for some constant $m$. Substituting $y = 98$ into $98 = 2\left(\frac{m}{z}\right)^2$ leads to $z^2 = \frac{2m^2}{98} = \frac{m^2}{49}$. Since with $z = \frac{7}{25}$, $x = 5$, we find $m$ and calculating for $z$ gives $z = \frac{1}{5}$.