Comparison of results across different $A_{\tau }\stackrel{\sim }{F}G$ ranges. Top panel: z_model vs. z_obs, same as Fig. 12, but with optimized f_IGM × H₀ values, where f_IGM × H₀ are $51.5_{-4.4}^{+4.4}$ km s⁻¹ Mpc⁻¹ and $63.2_{-5.2}^{+4.8}$ km s⁻¹ Mpc⁻¹ for the narrow and wide $A_{\tau }\stackrel{\sim }{F}G$ ranges, respectively. Middle panel: PDF of f_IGM × H₀ using the 30 FRB samples and our method. The solid black line corresponds to the wide $A_{\tau }\stackrel{\sim }{F}G$ range, and the dashed blue line corresponds to the narrow $A_{\tau }\stackrel{\sim }{F}G$ range. Bottom panel: PDF of H₀ for a given f_IGM = 0.85 ± 0.05 using the 30 FRB samples and our method.