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11.5 Exercise 5

Avg difficulty 2.5

Suppose that 18 respirators are to be apportioned to three hospitals based on their capacities. The Hamilton method is used to allocate the respirators in 2020, then to reallocate based on new capacities in 2021. The results are shown in the table below. How does this demonstrate the population paradox?

Hospital Capacity in 2020 Respirators in 2020 Change in Capacity Growth Rate = Change in Capacity Capacity in 2020\frac{\text{Change in Capacity }}{\text{Capacity in 2020}} Capacity in 2021 Respirators in 2021
A 825 9 57 57825≈0.0691=6.91%\frac{57}{825} \approx 0.0691 = 6.91 \% 882 9
B 613 7 13 13613≈0.0212=2.12%\frac{13}{613} \approx 0.0212 = 2.12 \% 626 6
C 239 2 3 3239=0.0126=1.26%\frac{3}{239} = 0.0126 = 1.26 \% 242 3

OpenStax Contemporary Mathematics, 11.5 Fairness in Apportionment Methods, exercise 5

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Adapted from Contemporary Mathematics © 2023 Rice University. Licensed under CC BY 4.0. Source: Contemporary Mathematics. Imported at revision fa7c1a1.