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The standard electroweak theory has two coupling constants $g$ and $g^\prime$. In this theory, the $W$ mass and $Z$ mass are given by $$M_W=\frac{1}{2}gv,~M_Z=\frac{1}{2}(g^2+g^{\prime 2})^{1/2}v$$ where $v$ is the vacuum expectation value of the neutral component of the Higgs doublet. But what was the prediction of the SM? It does not seem to be the absolute values of the W and Z masses because neither the values of $g$ and $g^\prime$ nor that of $v$ is a priori known.

In terms of Weinberg angle $\theta_W$, defined as $e=g\sin\theta_W$, $M_W=M_Z \cos\theta_W$ which trades another parameter in place of $g^\prime$. How is it possible to predict any one mass when the other mass and $\theta_W$ is also unknown?

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What you are stating is a "prediction" on the formalism that describes the electroweak interaction. The masses themselves had to be measured experimentally and the relations checked with the actual measured numbers. As you also state, no value could be extracted using the data at the time the model was proposed.

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