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Interactions in an ultracold boson-fermion mixture are often manifested by elastic collisions. In a mixture of a condensed Bose gas (BEC) and spin polarized degenerate Fermi gas (DFG), fermions can mediate spin-spin interactions between bosons, leading to an effective long-range magnetic interaction analogous to Ruderman-Kittel-Kasuya-Yosida [Phys. Rev. 96, 99 (1954); Prog. Theor. Phys. 16, 45 (1956); Phys. Rev. 106, 893 (1957)] interaction in solids. We used Ramsey spectroscopy of the hyperfine clock transition in a $^{87}\mathrm{Rb}$ BEC to measure the interaction mediated by a $^{40}\mathrm{K}$ DFG. By controlling the boson density we isolated the effect of mediated interactions from mean-field frequency shifts due to direct collision with fermions. We measured an increase of boson spin-spin interaction by a factor of $\ensuremath{\eta}=1.45\ifmmode\pm\else\textpm\fi{}0.0{5}^{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}0.1{3}^{\mathrm{syst}}$ in the presence of the DFG, providing clear evidence of spin-spin fermion mediated interaction. Decoherence in our system was dominated by inhomogeneous boson density shift, which increased significantly in the presence of the DFG, again indicating mediated interactions. We also measured a frequency shift due to boson-fermion interactions in accordance with a scattering length difference of ${a}_{b{f}_{2}}\ensuremath{-}{a}_{b{f}_{1}}=\ensuremath{-}5.36\ifmmode\pm\else\textpm\fi{}0.4{4}^{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}1.4{3}^{\mathrm{syst}}{a}_{0}$ between the clock-transition states, a first measurement beyond the low-energy elastic approximation [R. C\^ot\'e, A. Dalgarno, H. Wang, and W. C. Stwalley, Phys. Rev. A 57, R4118 (1998); A. Dalgarno and M. Rudge, Proc. R. Soc. A 286, 519 (1965)] in this mixture. This interaction can be tuned with a future use of a boson-fermion Feshbach resonance. Fermion-mediated interactions can potentially give rise to interesting new magnetic phases and extend the Bose-Hubbard model when the atoms are placed in an optical lattice.