Summary: We consider a \(d\)-dimensional well-formed weighted projective space \(\mathbb{P}(\overline{w})\) as a toric variety associated with a fan \(\varSigma(\overline{w})\) in \(N_{\overline{w}}\otimes\mathbb{R}\) whose 1-dimensional cones are spanned by primitive vectors \(v_0,v_1,\ldots,v_d\in N_{\overline{w}}\) generating a lattice \(N_{\overline{w}}\) and satisfying the linear relation \(\sum_iw_iv_i=0\). For any fixed dimension \(d\), there exist only finitely many weight vectors \(\overline{w}=(w_0,\ldots,w_d)\) such that \(\mathbb{P}(\overline{w})\) contains a quasi-smooth Calabi-Yau hypersurface \(X_w\) defined by a transverse weighted homogeneous polynomial \(W\) of degree \(w=\sum_{i=0}^dw_i\). Using a formula of Vafa for the orbifold Euler number \(\chi_{\mathrm{orb}}(X_w)\), we show that for any quasi-smooth Calabi-Yau hypersurface \(X_w\) the number \((-1)^{d-1}\chi_{\mathrm{orb}}(X_w)\) equals the stringy Euler number \(\chi_{\mathrm{str}}(X_{\overline{w}}^\ast)\) of Calabi-Yau compactifications \(X_{\overline{w}}^\ast\) of affine toric hypersurfaces \(Z_{\overline{w}}\) defined by non-degenerate Laurent polynomials \(f_{\overline{w}}\in\mathbb{C}[N_{\overline{w}}]\) with Newton polytope \(\mathrm{conv}(\{v_0,\ldots,v_d\})\). In the moduli space of Laurent polynomials \(f_{\overline{w}}\) there always exists a special point \(f_{\overline{w}}^0\) defining a mirror \(X_{\overline{w}}^\ast\) with a \(\mathbb{Z}/ w\mathbb{Z}\)-symmetry group such that \(X_{\overline{w}}^\ast\) is birational to a quotient of a Fermat hypersurface via a Shioda map.