= Solution
Choose basepoints and replace the spaces and map by <CW complex> models. If the hypothesis is stated for unreduced <represented homology theory>, the basepoint coefficient summands split naturally, so it also gives an isomorphism in reduced <represented homology theory>. Let $C$ be the homotopy <mapping cone> of $f$. Its reduced <represented homology theory> is zero in every degree, by the exact sequence of the corresponding <cofiber sequence of spectra>.
The <mapping cone> $C$ is a <simply connected space>: the mapping cylinder retracts onto $Z$, and coning off the path-connected space $Y$ adds no fundamental-group generators; the van Kampen calculation gives the quotient of $\pi_1Z$ by the image of $\pi_1Y$, which is trivial. Suppose some positive-degree integral <homology> group of $C$ were nonzero, and let $n$ be the least such degree. Then $n\ge2$. Starting with $\pi_1C=0$, the <Hurewicz theorem> shows inductively that $\pi_iC=0$ for $i<n$: at each first possible degree $i$, its <homotopy group> is identified with the zero group $H_i(C;\mathbb Z)$. It then identifies
$$
\pi_nC\cong H_n(C;\mathbb Z)\ne0.
$$
Applying the result of part (a) to the $(n-1)$-connected <based space> $C$ gives $\widetilde E_n(C)\cong\pi_nC\ne0$, contradicting the vanishing of its <represented homology theory>.
Thus $C$ is integrally acyclic, and the ordinary <homology> exact sequence says that $f$ is an integral homology isomorphism. The allowed <homological Whitehead theorem> for <simply connected spaces> makes the map of <CW complex> models a homotopy equivalence. Returning to the original spaces therefore gives
$$
\boxed{f\text{ is a weak homotopy equivalence}.}
$$
The simply connected hypothesis is used both in the first-nonzero-degree <Hurewicz theorem> argument and in the final <homological Whitehead theorem>.
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