11.4.0: Exercises
- Page ID
- 171753
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For the following exercises, use the standard quotas given in the table below.
| State A | State B | State C | State D | State E | State F | Total Seats | |
|---|---|---|---|---|---|---|---|
| Scenario X | 17.63 | 26.62 | 10.81 | 16.01 | 13.69 | 15.24 | 100 |
| Scenario Y | 12.37 | 7.59 | 71.71 | 6.75 | 5.76 | 20.81 | 125 |
| Scenario Z | 3.53 | 31.56 | 2.95 | 5.12 | 9.84 | NA | 53 |
Round the standard quota for each state in Scenario X using traditional rounding. Find the sum of the modified quotas. What is the difference between the sum and the house size?
Round the standard quota for each state in Scenario Y using traditional rounding. Find the sum of the modified quotas. What is the difference between the sum and the house size?
Round the standard quota for each state in Scenario Z using traditional rounding. Find the sum of the modified quotas. What is the difference between the sum and the house size?
Find the lower quota for each state in Scenario Y. If each state is allocated its lower quota, how many seats remain to be allocated?
Find the lower quota for each state in Scenario X. If each state is allocated its lower quota, how many seats remain to be allocated?
Find the lower quota for each state in Scenario Z. If each state is allocated its lower quota, how many seats remain to be allocated?
Find the upper quota for each state in Scenario X and determine how much the sum of the upper quotas exceeds the house size.
Find the upper quota for each state in Scenario Y and determine how much the sum of the upper quotas exceeds the house size.
Find the upper quota for each state in Scenario Z and determine how much the sum of the upper quotas exceeds the house size.
Determine the Hamilton apportionment for Scenario Y.
Determine the Hamilton apportionment for Scenario X.
Determine the Hamilton apportionment for Scenario Z.
For the following exercises, use the information in the table below, which gives standard and final quotas for Methods X, Y, and Z.
| State A | State B | State C | State D | State E | |
|---|---|---|---|---|---|
| Standard Quota | 1.67 | 3.33 | 5.00 | 6.67 | 8.33 |
| Method X | 2 | 2 | 5 | 7 | 9 |
| Method Y | 1 | 3 | 5 | 7 | 9 |
| Method Z | 1 | 3 | 5 | 6 | 10 |
Does the apportionment resulting from method X satisfy the quota rule? Why or why not?
Does the apportionment resulting from method Z satisfy the quota rule? Why or why not?
Does the apportionment resulting from method Y satisfy the quota rule? Why or why not?
In the movie Black Panther, the hero lives in the fictional country of Wakanda. Imagine that 111 Vibranium artifacts must be distributed among the fortress cities, or birnin, of Wakanda based on the population of each birnin. Use the population and standard quota information in the table below for the following exercises.
| Birnin Djata (D) | Birnin T'Chaka (T) | Birnin Zana (Z) | Birnin S'Yan (S) | Birnin Bashenga (B) | Birnin Azzaria (A) | Total | |
|---|---|---|---|---|---|---|---|
| Residents | 26,000 | 57,000 | 27,000 | 18,000 | 64,000 | 45,000 | 237,000 |
| Standard Quota | 12.18 | 26.70 | 12.65 | 8.43 | 29.98 | 21.08 | 111 |
Modify the standard quota for each state using traditional rounding. Find the sum of the modified quotas. What is the difference between the sum and the house size?
Find the standard lower quota for each state. If each state is allocated its lower quota, how many seats remain to be allocated?
Find the standard upper quota for each state and determine how much the sum of the upper quotas exceeds the house size.
Use the Hamilton method to apportion the artifacts.
Find the modified lower quota for each state using a modified divisor of 2,000. Is the sum of the modified quotas too high, too low, or equal to the house size?
Find the modified lower quota for each state using a modified divisor of 2,100. Is the sum of the modified quotas too high, too low, or equal to the house size?
Use the Jefferson method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.
Does the Jefferson method result in an apportionment that satisfies or violates the quota rule in this scenario?
Find the modified upper quota for each state using a modified divisor of 2,250. Is the sum of the modified quotas too high, too low, or equal to the house size?
Find the modified upper quota for each state using a modified divisor of 2,150. Is the sum of the modified quotas too high, too low, or equal to the house size?
Use the Adams method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.
Does the Adams method result in an apportionment that satisfies or violates the quota rule in this scenario?
Which method of apportionment, Jefferson or Adams, is a resident of Birnin T'Chaka likely to prefer? Justify your answer.
Use the Webster method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so,indicate the value of the modified divisor.
Does the Webster method result in an apportionment that satisfies or violates the quota rule in this scenario?
Which of the four methods of apportionment from this section (Hamilton, Jefferson, Adams, or Webster) are the residents of Birnin S'Yan likely to prefer? Justify your answer.
Children from five families—the Chorro family, the Eswaran family, the Javernick family, the Lahde family, and the Stolly family—joined a town Easter egg hunt. When they returned with their baskets, they had 827 eggs! They decided to share their eggs amongst the families based on the number of children in each family. Use the population and standard quota information in the table below for the following exercises.
| (C) Chorro | (E) Eswaran | (J) Javernick | (L) Lahde | (S) Stolly | Total | |
|---|---|---|---|---|---|---|
| Children | 3 | 2 | 4 | 1 | 5 | 15 |
| Standard Quota | 155.04 | 103.36 | 206.72 | 103.36 | 258.40 | 827 |
Modify the standard quota for each state using traditional rounding. Find the sum of the modified quotas. What is the difference between the sum and the house size?
Find the standard lower quota for each state. If each state is allocated its lower quota, how many seats remain to be allocated?
Find the standard upper quota for each state, and determine how much the sum of the upper quotas exceeds the house size.
Use the Hamilton method to apportion the Easter eggs.
Find the modified lower quota for each state using a modified divisor of 0.01800. Is the sum of the modified quotas too high, too low, or equal to the house size?
Find the modified lower quota for each state using a modified divisor of 0.01810. Is the sum of the modified quotas too high, too low, or equal to the house size?
Use the Jefferson method to apportion the Easter eggs. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.
Does the Jefferson method result in an apportionment that satisfies or violates the quota rule in this scenario?
Find the modified upper quota for each state using a modified divisor of 0.0182. Is the sum of the modified quotas too high, too low, or equal to the house size?
Find the modified upper quota for each state using a modified divisor of 0.01816. Is the sum of the modified quotas too high, too low, or equal to the house size?
Use the Adams method to apportion the Easter eggs. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.
Does the Adams method result in an apportionment that satisfies or violates the quota rule in this scenario?
Use the Webster method to apportion the Easter eggs. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.
Does the Webster method result in an apportionment that satisfies or violates the quota rule in this scenario?
For the following exercises, use this information: Suppose that the State of Delaware received 2,000 packs of COVID-19 vaccines, with ten doses per pack. These (unopened) packs must be distributed to the three counties based on total population. Use the population information in the table below to determine how many packs of vaccine will be distributed to each county based on the given apportionment method.
| (N) New Castle | (K) Kent | (S) Sussex | |
|---|---|---|---|
| Residents | 558,753 | 180,786 | 234,225 |
Hamilton’s Method
Jefferson’s Method
Adams's Method
Webster’s Method
Notice that the apportionments found in questions 46, 47, 48, and 49 all satisfy the quota rule. Does this contradict the statement, “The Jefferson, Adams, and Webster methods of apportionment all violate the quota rule”? Why or why not?