Math / Data & Statistics

Question2. Refer to the following preference schedule to answer parts (a) - (f). Note: See attached page for additional copies of the table. ( $36 \mathrm{pts}-4$ pts for each part) \begin{tabular}{|r|c|c|c|c|} \hline First & A & B & D & C \\ \hline Second & B & D & C & B \\ \hline Third & C & A & B & A \\ \hline Forth & D & C & A & D \\ \hline Number of Voters & 30 & 20 & 25 & 10 \\ \hline \end{tabular} a. How many votes were cast? $\qquad$ b. How many votes are needed to receive a majority? $\qquad$ c. Find the plurality winner: $\qquad$ d. Did the plurality winner also receive a majority? $\qquad$ Show work to support the selection process below. e. Find the winner by a single runoff of the top two candidates: $\qquad$ Show work to support the selection process below. f. Find the winner by a sequential runoff: $\qquad$ Show work to support the selection process below. g. Find the winner by a Borda count: $\qquad$ Show work to support the selection process below. h. Find the winner, if any, by the method of pairwise comparisons: $\qquad$ Show work to support the selection process below.

Studdy Solution

STEP 1

1. The preference schedule is given as a table with candidates A, B, C, and D.
2. The number of voters for each preference order is provided.
3. The tasks involve determining winners using different voting methods.

STEP 2

1. Calculate the total number of votes cast.
2. Determine the number of votes needed for a majority.
3. Identify the plurality winner.
4. Check if the plurality winner also received a majority.
5. Determine the winner by a single runoff of the top two candidates.
6. Determine the winner by a sequential runoff.
7. Determine the winner by a Borda count.
8. Determine the winner by the method of pairwise comparisons.

STEP 3

Calculate the total number of votes cast by summing the number of voters for each preference order:
$30 + 20 + 25 + 10 = 85$

STEP 4

Calculate the number of votes needed to receive a majority. A majority is more than half of the total votes:
$\text{Majority} = \frac{85}{2} = 42.5$
Since a candidate cannot receive a fraction of a vote, round up to the next whole number:
$43$

STEP 5

Identify the plurality winner by finding the candidate with the most first-place votes:
- A: 30 votes - B: 20 votes - D: 25 votes - C: 10 votes
The plurality winner is candidate A with 30 votes.

STEP 6

Check if the plurality winner (A) also received a majority:
Candidate A received 30 votes, which is less than the majority of 43 votes.

STEP 7

Determine the winner by a single runoff of the top two candidates (A and D):
- A: 30 votes - D: 25 votes
In a runoff, consider the second preferences of voters who did not vote for A or D initially:
- 20 voters prefer B first, then D - 10 voters prefer C first, then B
Redistribute these votes:
- A: 30 votes - D: 25 + 20 = 45 votes
The winner by a single runoff is candidate D.

STEP 8

Determine the winner by a sequential runoff:
1. Eliminate the candidate with the fewest first-place votes (C with 10 votes).
2. Redistribute C's votes based on second preferences (to B).
3. New totals: A: 30, B: 20 + 10 = 30, D: 25
4. Eliminate B (tied with A but fewer initial votes).
5. Redistribute B's votes based on second preferences (to D).
6. New totals: A: 30, D: 25 + 20 = 45
The winner by a sequential runoff is candidate D.

STEP 9

Determine the winner by a Borda count:
Assign points based on rank (3 for first, 2 for second, 1 for third, 0 for fourth):
- A: $30 \times 3 + 20 \times 2 + 25 \times 1 + 10 \times 0 = 90 + 40 + 25 + 0 = 155$ - B: $30 \times 2 + 20 \times 0 + 25 \times 1 + 10 \times 3 = 60 + 0 + 25 + 30 = 115$ - C: $30 \times 1 + 20 \times 3 + 25 \times 0 + 10 \times 2 = 30 + 60 + 0 + 20 = 110$ - D: $30 \times 0 + 20 \times 1 + 25 \times 3 + 10 \times 1 = 0 + 20 + 75 + 10 = 105$
The winner by Borda count is candidate A.

STEP 10

Determine the winner by the method of pairwise comparisons:
Compare each pair of candidates:
- A vs. B: A wins (30 vs. 20) - A vs. C: A wins (30 vs. 10) - A vs. D: D wins (45 vs. 30) - B vs. C: B wins (20 vs. 10) - B vs. D: D wins (45 vs. 20) - C vs. D: D wins (45 vs. 10)
D wins the most pairwise comparisons.
The winner by the method of pairwise comparisons is candidate D.

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Studdy Solution

STEP 1

1. The preference schedule is given as a table with candidates A, B, C, and D.
2. The number of voters for each preference order is provided.
3. The tasks involve determining winners using different voting methods.

STEP 2

STEP 3

Calculate the total number of votes cast by summing the number of voters for each preference order:
$30 + 20 + 25 + 10 = 85$

STEP 4

Calculate the number of votes needed to receive a majority. A majority is more than half of the total votes:
$\text{Majority} = \frac{85}{2} = 42.5$
Since a candidate cannot receive a fraction of a vote, round up to the next whole number:
$43$

STEP 5

Identify the plurality winner by finding the candidate with the most first-place votes:
- A: 30 votes - B: 20 votes - D: 25 votes - C: 10 votes
The plurality winner is candidate A with 30 votes.

STEP 6

Check if the plurality winner (A) also received a majority:
Candidate A received 30 votes, which is less than the majority of 43 votes.

STEP 7

STEP 8

STEP 9

STEP 10

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Studdy Solution

STEP 1

1. The preference schedule is given as a table with candidates A, B, C, and D.2. The number of voters for each preference order is provided.3. The tasks involve determining winners using different voting methods.

STEP 2

STEP 3

Calculate the total number of votes cast by summing the number of voters for each preference order:30+20+25+10=85 30 + 20 + 25 + 10 = 85 30+20+25+10=85

STEP 4

Calculate the number of votes needed to receive a majority. A majority is more than half of the total votes:Majority=852=42.5 \text{Majority} = \frac{85}{2} = 42.5 Majority=285​=42.5Since a candidate cannot receive a fraction of a vote, round up to the next whole number:43 43 43

STEP 5

Identify the plurality winner by finding the candidate with the most first-place votes:- A: 30 votes - B: 20 votes - D: 25 votes - C: 10 votesThe plurality winner is candidate A with 30 votes.

STEP 6

Check if the plurality winner (A) also received a majority:Candidate A received 30 votes, which is less than the majority of 43 votes.

STEP 7

STEP 8

STEP 9

STEP 10

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The preference schedule is given as a table with candidates A, B, C, and D.
2. The number of voters for each preference order is provided.
3. The tasks involve determining winners using different voting methods.

Calculate the total number of votes cast by summing the number of voters for each preference order:
$30 + 20 + 25 + 10 = 85$

Calculate the number of votes needed to receive a majority. A majority is more than half of the total votes:
$\text{Majority} = \frac{85}{2} = 42.5$
Since a candidate cannot receive a fraction of a vote, round up to the next whole number:
$43$

Identify the plurality winner by finding the candidate with the most first-place votes:
- A: 30 votes - B: 20 votes - D: 25 votes - C: 10 votes
The plurality winner is candidate A with 30 votes.

Check if the plurality winner (A) also received a majority:
Candidate A received 30 votes, which is less than the majority of 43 votes.