11.5.0: Exercises
- Page ID
- 171754
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In the following exercises, determine whether the scenario violates the Alabama paradox. Justify your answer.
A company with an office in each of four cities must distribute 145 new Chromebooks to the four offices. It is determined that Office A will receive 42, Office B will receive 17, Office C will receive 35, and Office D will receive 51. At the last minute, it is discovered that there are 146 Chromebooks. When they are reapportioned, Office A receives 42, Office B receives 16, Office C receives 36, and Office D receives 52.
A county with three towns has 30 garbage trucks to apportion. Attenborough receives 6 trucks, Breckenridge receives 8 trucks, and Cabbotsville receives 16 trucks, in proportion to their populations. When one additional truck is purchased, the reallocation results in 5 trucks for Attenborough, 9 for Breckenridge, and 8 for Cabbotsville though there has been no change in populations.
A group of 60 dentists work for a company that runs five offices. The dentists have been apportioned to the offices by the number of patients. Office A receives 14 dentists, Office B receives 11, Office C receives 11, Office D receives 12, and Office E receives 12. When a new dentist joins the group, the new apportionment gives the following: A 14, B 11, C 11, D 12, and E 13.
In the following exercises, determine whether the scenario violates the population paradox. Justify your answer.
A company with locations in three cities plans to give 200 achievement awards that shall be apportioned to the three cities by population. City A with 9,150 employees is allotted 61 awards, City B with 6,040 employees is allotted 40 awards, and City C with 14,810 employees is allotted 99 awards to distribute. Then it is discovered that the number of employees is out of date. The awards are reallocated based on the new populations: City A with 9,180; City B with 6,040; and City C with 14,930. It turns out that the apportionment remains the same.
A soccer club must apportion soccer balls to the teams among four age brackets based on the number of teams in each bracket. Bracket U8 receives 32 balls, U12 receives 46 balls, U15 receives 29 balls, and U18 receives 25 balls. Mid-season, the balls are reapportioned. U8 has decreased by 10 percent, U12 has increased by 20 percent, U15 remains the same, and U18 has increased by 10 percent. The reapportionment gives 33 balls to U8, 46 balls to U12, 29 balls to U15, and 14 balls to U18.
READ (Reading Education Assistance Dogs) trains and certifies therapy animals to serve in classrooms and help children develop a love of reading in a low-stress environment. Suppose that 20 therapy teams are apportioned to three schools based on their populations. When the population of School A decreases by 10 percent, the population of School B increases by 10 percent, and the population of School C remains constant, the apportionment remains the same.
In the following exercises, determine whether the scenario violates the new-states paradox. Justify your answer.
A charity organization has 851 volunteers in Country A and 3449 volunteers in Country B. There are 43 lead organizers that must be apportioned to the two locations. Country A receives 9 while Country B receives 34. When the operations are expanded to Country C with 725 volunteers, 7 new lead organizers are added to the team. When the lead organizers are reapportioned, Country A receives 9, Country B receives 34, and Country C receives 7 lead organizers.
A country has two states. There are 22 seats in the legislature. State A has 4 seats, and State B has 18 seats. When State C joins the union, the number of representatives is increased by five. Under the new apportionment, State A receives 3 seats, State B receives 19 seats, and State C receives 5 seats.
In the Garunga Solar System, there are four inhabited planets. Three of the planets are members of the United Association of Garungan Planets (UAGP). The UAGP has 67 seats. Planet Angluertka has 40 seats, planet Bangluertka has 27 seats, and planet Clangluertka has 13 seats. When planet Danggluertka decides to join the UAGP, 14 seats are added to accommodate them proportionately. The new allocation of seats gives 40 seats to Angluertka, 26 seats to Bangluertka, 14 seats to Clangluertka, and 14 seats to Danggluertka.
In the following exercises, use the Hamilton method of apportionment to answer the questions.
When the number of seats changed from 147 to 148, the standard quotas changed from A 44.24, B 17.35, C 37.12, and D 48.29 to A 45.54, B 17.47, C 37.37, and D 48.62.
- How did the increase in seats impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
When the number of seats changed from 126 to 127, the standard quotas changed from A 9.57, B 29.49, C 33.89, D 28.43, and E 24.61 to A 9.65, B 29.72, C 34.16, D 38.66, and E 24.81.
- How did the increase in seats impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
When the number of seats changed from 25 to 26, the standard quotas changed from A 2.21, B 5.25, C 11.27, and D 6.27 to A 2.30, B 5.46, C 11.72, and D 6.52.
- How did the increase in seats impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
When the number of seats changed from 25 to 26, the standard quotas changed from A 2.43, B 5.42, and C 8.15 to A 2.46, B 5.47, and C 8.07.
- How did the increase in seats impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
The house size is 18. When the population of State A increases by 11.76 percent, State B increases by 16.22 percent, and State C increases by 12.18 percent, the standard quotas change from A 2.46, B 6.00, and C 9.53 to A 2.41, B 6.25, and C 9.34.
- How did the change in populations impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
The house size is 100. When the population of State A increases by 20 percent, State B increases by 10 percent, State C increases by 30 percent, and the populations of States D, E, and F remain the same. The standard quotas change from A 12.50, B 25.00, C 9.38, D 18.75, E 12.50 and F 21.88 to A 13.91, B 25.51, C 11.30, D 17.39, E 11.59, and F 20.29.
- How did the change in populations impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
The house size is 17. When the population of State A increases by 12.73 percent, State B increases by 15.63 percent, and State C increases by 12.90 percent, the standard quotas change from A 2.53, B 5.90, and C 8.57 to A 2.51, B 5.99, and C 8.50.
- How did the change in populations impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
The house size is 38 seats. When the population of State A increases by 6.60 percent, State B increases by 1.44 percent, and State C increases by 1.40 percent, the standard quotas change from A 18.82, B 14.65, and C 4.53 to A 19.59, B 14.29, and C 4.41.
- How did the change in populations impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
The house size is 24 seats. When the population of State A increases by 28 percent, State B increases by 26 percent, and State C increases by 15 percent, the standard quotas change from A 3.38, B 6.32, and C 14.30 to A 3.63, B 6.67, and C 13.71.
- How did the change in populations impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
The house size was 60. There were three states with standard quotas of A 4.18, B 15.38, and C 40.44. A fourth state was annexed, and the house size was increased to 65. The new standard quotas are A 4.17, B 15.33, C 40.32, and D 5.18.
- How did the additional state impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
The house size was 50. There were three states with standard quotas of A 9.41, B 24.42, and C 16.17. A fourth state was annexed, and the house size was increased to 66. The new standard quotas are A 9.36, B 24.30, C 16.09, and D 16.25.
- How did the additional state impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
The house size was 27. There were three states with standard quotas of A 6.39, B 11.40, and C 9.21. A fourth state was annexed, and the house size was increased to 35. The new standard quotas are A 6.38, B 11.37, C 9.19, and D 8.06.
- How did the additional state impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
The house size was 100. There were three states with standard quotas of A 26.09, B 30.43, and C 43.48. A fourth state was annexed, and the house size was increased to 122. The new standard quotas are A 26.14, B 30.50, C 43.57, and D 21.78.
- How did the additional state impact the apportionment?
- Is this apportionment an example of a paradox? Justify your answer.
In the following exercises, use the information in the table below.
| State | A | B | C | D | E | F | G | H | I | J | K | L | P | Q | R |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Population | 624 | 1,219 | 979 | 3,462 | 7,470 | 4,264 | 5,300 | 263 | 809 | 931 | 781 | 676 | 150 | 250 | 350 |
| Original House Size | 38 | 204 | 126 | 50 | |||||||||||
| Updated House Size | 39 | 205 | 127 | 51 | |||||||||||
Consider States A, B, and C.
- Determine the apportionment for States A, B, and C with the original house size using the Hamilton method.
- Determine the apportionment for States A, B, and C with the updated house size using the Hamilton method.
- Does the change in the house size and use of the Hamilton method cause the Alabama paradox? Explain your reasoning.
Consider States D, E, F, and G.
- Determine the apportionment for States D, E, F, and G with the original house size using the Hamilton method.
- Determine the apportionment for States D, E, F, and G with the updated house size using the Hamilton method.
- Does the change in the house size and use of the Hamilton method cause the Alabama paradox? Explain your reasoning.
Consider States H, I, J, K, and L.
- Determine the apportionment for States H, I, J, K, and L with the original house size using the Hamilton method.
- Determine the apportionment for States H, I, J, K, and L with the updated house size using the Hamilton method.
- Does the change in the house size and use of the Hamilton method cause the Alabama paradox? Explain your reasoning.
Consider States P, Q, and R.
- Determine the apportionment for States P, Q, and R with the original house size using the Hamilton method.
- Determine the apportionment for States P, Q, and R with the updated house size using the Hamilton method.
- Does the change in the house size and use of the Hamilton method cause the Alabama paradox? Explain your reasoning.
In the following exercises, use the information in the table below.
| State | Original Population | Updated Population | Population Growth Rate | House Size |
|---|---|---|---|---|
| A | 889 | 958 | 7.76% | |
| B | 674 | 692 | 2.67% | 42 |
| C | 237 | 243 | 2.53% | |
| D | 12,032 | 14,124 | \(\bf{\it{d\%}}\) | |
| E | 10,789 | 9,726 | \(\bf{\it{e\%}}\) | 135 |
| F | 995 | 2,304 | \(\bf{\it{f\,\%}}\) | |
| G | 901 | 1,156 | 28.3% | |
| H | 1,683 | 2,125 | 26.3% | 24 |
| I | 3,808 | 4,369 | 14.7% | |
| X | 56 | 63 | \(\bf{\it{x\%}}\) | |
| Y | 125 | 141 | \(\bf{\it{y\%}}\) | 16 |
| Z | 182 | 213 | \(\bf{\it{z\%}}\) | |
| P | 6,534 | 6,534 | \(p\%\) | |
| Q | 7,832 | 7,810 | \(q\%\) | 40 |
| R | 13,959 | 13,992 | \(r\%\) | |
| S | 20,515 | 21,164 | \(s\%\) | |
Calculate the population growth rates \(d\), \(e\), and \(f\) for States D, E, and F. Give answer as a percentage rounded to one decimal place.
Calculate the population growth rates \(p\), \(q\), \(r\), and \(s\) for States P, Q, R, and S. Give answer as a percentage rounded to one decimal place.
Calculate the population growth rates \(x\), \(y\), and \(z\) for States X, Y, and Z. Give answer as a percentage rounded to one decimal place.
Consider States A, B, and C.
- Determine the Hamilton apportionment for States A, B, and C with the original population.
- Determine the Hamilton apportionment for States A, B, and C with the updated population.
- Does the increase in population of States A, B, and C from the original population to the updated population and the use of the Hamilton method cause the population paradox? Explain your reasoning.
Consider States D, E, and F.
- Determine the Hamilton apportionment for States D, E, and F with the original population.
- Determine the Hamilton apportionment for States D, E, and F with the updated population.
- Does the increase in population and the use of the Hamilton method cause the population paradox? Explain your reasoning.
Consider States G, H, and I.
- Determine the Hamilton apportionment for States G, H, and I with the original population.
- Determine the Hamilton apportionment for States G, H, and I with the updated population.
- Does the increase in population and the use of the Hamilton method cause the population paradox? Explain your reasoning.
Consider States X, Y, and Z.
- Determine the Hamilton apportionment for States X, Y, and Z with the original population.
- Determine the Hamilton apportionment for States X, Y, and Z with the updated population.
- Does the increase in population and the use of the Hamilton method cause the population paradox? Explain your reasoning.
Consider States P, Q, R, and S.
- Determine the Hamilton apportionment for States P, Q, R, and S with the original population.
- Determine the Hamilton apportionment for States P, Q, R, and S with the updated population.
- Does the increase in population and the use of the Hamilton method cause the population paradox? Explain your reasoning.
In the following exercises, use the information in the table below.
| State | Population | Original House Size | New House Size |
|---|---|---|---|
| A | 627 | ||
| B | 1,287 | 25 | |
| C | 973 | 32 | |
| D | 815 | ||
| E | 520 | ||
| F | 1,510 | 50 | |
| G | 1,060 | \(\bf{\it{h}}\) | |
| H | 950 | ||
| P | 1,222 | 100 | |
| Q | 473 | ||
| R | 225 | \(\bf{\it{r}}\) | |
| K | 1,688 | ||
| L | 7,912 | 48 | |
| M | 1,448 | \(\bf{\it{m}}\) | |
| T | 150 | ||
| U | 250 | ||
| V | 350 | 50 | \(\bf{\it{w}}\) |
| W | 450 |
Consider States A, B, and C.
- Calculate the standard divisor based on the original house size.
- Use the Hamilton method to apportion the seats.
Consider States E, F, and G.
- Calculate the standard divisor based on the original house size.
- Use the Hamilton method to apportion the seats.
Consider States P and Q.
- Calculate the standard divisor based on the original house size.
- Use the Hamilton method to apportion the seats.
Consider States K and L.
- Calculate the standard divisor based on the original house size.
- Use the Hamilton method to apportion the seats.
Suppose that States A, B, and C annex State D and increase the house size proportionately.
- Calculate the standard divisor based on the new house size.
- Use the Hamilton method to reapportion the seats.
- Does the new-states paradox occur?
Suppose that States E, F, and G annex State H and increase the house size proportionately.
- Determine the new house size, \(h\), that is necessary.
- Calculate the standard divisor based on the new house size.
- Use the Hamilton method to reapportion the seats.
- Does the new-states paradox occur?
Suppose that States P and Q annex State R and increase the house size proportionately.
- Determine the new house size, \(r\), that is necessary.
- Calculate the standard divisor based on the new house size.
- Use the Hamilton method to reapportion the seats.
- Does the new-states paradox occur?
Suppose that States K and L annex State M and increase the house size proportionately.
- Determine the new house size, \(m\), that is necessary.
- Calculate the standard divisor based on the new house size.
- Use the Hamilton method to reapportion the seats.
- Does the new-states paradox occur?
Suppose that States T, U, and V annex State W and increase the house size proportionately.
- Calculate the standard divisor based on the original house size and only States T, U, and V.
- Use the Hamilton method to apportion the seats to T, U, and V.
- Determine the new house size when State W is annexed.
- Calculate the standard divisor based on the new house size.
- Use the Hamilton method to reapportion the seats.
- Does the new-states paradox occur? (Refer to part b.)
Suppose 24 seats are apportioned to States A, B, and C with populations of 16, 15, and 125 respectively. Then the populations of States A, B, and C change to 17, 15, and 126 respectively.
- Demonstrate that the population paradox occurs when the Hamilton method is used.
- Determine whether the population paradox occurs when the Webster method is used. Justify your answer.
Suppose that 10 seats are apportioned to States A, B, and C with populations 6, 6, and 2 respectively. Then the number of seats is increased to 11. Demonstrate that the Alabama paradox occurs when the Hamilton method is used.