11.6.6: Chapter Test
- Page ID
- 129668
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)| Number of Ballots | 12 | 17 | 15 | 13 |
|---|---|---|---|---|
| Option A | 1 | 2 | 4 | 3 |
| Option B | 3 | 1 | 3 | 2 |
| Option C | 4 | 3 | 1 | 4 |
| Option D | 2 | 4 | 2 | 1 |
Determine the winner of the election by plurality.
Determine the Borda scores for each candidate to determine the winner by Borda count method.
Create and analyze a pairwise comparison matrix based on the preference summary to determine the winner of the election by pairwise comparison.
From the table below use ranked-choice voting to determine the winner of the election.
| Number of Ballots | 28 | 5 | 30 | 5 | 16 | 16 |
|---|---|---|---|---|---|---|
| Option L | 3 | 2 | 1 | 1 | 2 | 3 |
| Option R | 1 | 1 | 3 | 2 | 3 | 2 |
| Option E | 2 | 3 | 2 | 3 | 1 | 1 |
For the following exercises, identify which fairness criteria, if any, are violated by characteristics of the described voter profile. Explain your reasoning
In a Borda count election, the candidates have the following Borda scores: A 1345, B 1260, C 685. Candidate B received 51% of the first-place rankings.
In a plurality election, the candidates have the following percentages of first place votes: A 25, B 21, C 30, D 24. The pairwise matchup points for the same voter profiles would have been A 3, B 0, C 2, D 2.
For the following exercises, use the table below.
| Number of Ballots | 13 | 14 | 11 | 12 |
|---|---|---|---|---|
| Option A | 2 | 1 | 3 | 3 |
| Option B | 3 | 2 | 4 | 1 |
| Option C | 4 | 4 | 1 | 2 |
| Option D | 1 | 3 | 3 | 4 |
Determine the winner by ranked-choice voting if two of the voters in the second column up-rank the original winner. Refer to Question 4. Which fairness criterion, if any, is violated?
Determine the winner by ranked-choice voting if candidate R is removed from the election. Refer to Question 4. Which fairness criterion, if any, is violated?
For the following exercises, use this information: The incorporated town of Orange Grove consists of two subdivisions: The Oaks with 1,254 residents, and The Villages with 10,746 residents. A council with 100 members supervises the municipality's operations with representation proportionate to the number of residents.
Identify the states, the seats, and the state population (the basis for apportionment) in the given scenario.
Determine the standard divisor for the apportionment
Determine each state's standard quota rounded to two decimal places.
For the following exercises, use this information: Air Force administration wanted to distribute 27 aircraft across 6 bases based on the number of qualified pilots stationed at those bases. The standard quota is 2.2963. The standard quotas for each base are listed in the table below.
| Air Force Base | (A) Alpha | (B) Bravo | (C) Charlie | (D) Delta | (E) Echo | (F) Foxtrot |
|---|---|---|---|---|---|---|
| Pilots | 13 | 12 | 5 | 16 | 7 | 9 |
| Standard Quota | 5.66 | 5.23 | 2.18 | 6.97 | 3.05 | 3.92 |
Determine the states' lower quotas and the states' upper quotas.
Use Adams's method to apportion the aircraft.
Use Jefferson's method to apportion the aircraft.
The apportionment of 616 schools to 5 Hawaiian counties by various methods is displayed in the table below.
| County | Hawaii | Honolulu | Kalawao | Kauai | Maui |
|---|---|---|---|---|---|
| Lower Quota | 87 | 424 | 0 | 31 | 72 |
| Upper Quota | 88 | 425 | 1 | 32 | 73 |
| Jefferson | 87 | 425 | 1 | 31 | 72 |
| Adams | 88 | 422 | 1 | 32 | 73 |
| Webster | 87 | 424 | 1 | 31 | 73 |
Apportionment by which methods, if any, fail to satisfy the quota rule? Explain your reasoning. For the following exercises, use this information: The incorporated town of Orange Grove consists of two subdivisions: The Oaks with 1,254 residents, and The Villages with 10,746 residents. A council with 100 members supervises the municipality's operations. The Hamilton method was used to apportion the council seats. The Oaks has 10 seats on the council, while The Villages has 90 seats. The council votes to annex an unincorporated subdivision called The Lakes with a population of 630. They plan to increase the size of the council to maintain the ratio of seats to residents such that the new council will have 100 seats plus the number of seats given to The Lakes.
What is the standard divisor from the original apportionment?
What is the new house size?
Use the Hamilton method to reapportion the seats.
Is the reapportionment an example of the new-states paradox? If so, how?
For the following exercises, use this information: determine whether the reapportionment violates the Alabama paradox, the population paradox, or neither. Justify your answer.
States A, B, C, and D received 21, 25, 26, and 28 seats respectively. When the population remains the same, but house size is increased, the reapportionment is A 20, B 26, C 27, and D 29.
States A, B, C, and D received 21, 25, 26, and 28 seats respectively. When the house size remains the same, the population of state A increased, the population of state B decreased, and the populations of states C and D remained the same, the reapportionment is A 20, B 26, C 26, and D 28.