Step 1: Understanding the Concept:
Hybridization depends on the number of sigma bonds and lone pairs. A lone pair that is delocalized via resonance is considered to be in a p-orbital, lowering the hybridization index.
Step 2: Detailed Explanation:
1. Phenyl acetate: The oxygen lone pair is in resonance with the carbonyl group. $\rightarrow$ \(sp^2\).
2. Vinyl ether: The oxygen lone pair is in resonance with the adjacent double bond. $\rightarrow$ \(sp^2\).
3. Dioxolane: The oxygen has two sigma bonds and two non-delocalized lone pairs in a saturated ring. $\rightarrow$ \(sp^3\).
4. Furan: The oxygen is part of an aromatic ring and must contribute a lone pair to the \(\pi\) system. $\rightarrow$ \(sp^2\).
Step 3: Final Answer:
The sequence is \(sp^2, sp^2, sp^3, sp^2\).