It’s comparable to what we have seen and calculated in this course. The formula for the splitted energy levels due to the Zeemann-effect is
(- µ dot B) with µ = -g(µ_B/hbar)S. Note that B, µ and S are all vectors. If we apply B according to the z-direction, we have vec(B)=B*vec(z) and thus the dot product will result in a [vec(S) dot vec(z)], and we take S quantized along the z-axis, which will result in (m_z hbar). Hope this helps!