Use the following distribution of 33 voters at eight different positions over the interval [0, 1] to answer Questions \[\begin{array} { | l | c | c | c | c | c | c | c | c | } \hline \text { Position } & 0.1 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline \begin{array} { l } \text { Number of } \\ \text { voters } \end{array} & 1 & 4 & 6 & 6 & 6 & 2 & 5 & 3 \\ \hline \end{array}\] -What is the median of the distribution?

A) 0.5 B) 0.55 C) 0.56 D) 0.6 A) 0.5 B) 0.55 C) 0.56 D) 0.6

Use the following distribution of 33 voters at eight different positions over the interval [0, 1] to answer Questions \[\begin{array} { | l | c | c | c | c | c | c | c | c | } \hline \text { Position } & 0.1 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline \begin{array} { l } \text { Number of } \\ \text { voters } \end{array} & 1 & 4 & 6 & 6 & 6 & 2 & 5 & 3 \\ \hline \end{array}\] -For two candidates A and B with distinct policy positions 0.3 and 0.8, respectively, find the number of votes for the candidate A based on voter's ideal positions.

A) 16 B) 17 C) 23 D) 25 A) 16 B) 17 C) 23 D) 25

Use the following distribution of 32 voters at eight different positions over the interval [0, 1] to answer the Questions \[\begin{array} { | l | c | c | c | c | c | c | c | c | } \hline \text { Position } & 0.1 & 0.2 & 0.3 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline \begin{array} { l } \text { Number of } \\ \text { voters } \end{array} & 6 & 5 & 5 & 3 & 4 & 6 & 2 & 1 \\ \hline \end{array}\] -For two candidates A and B with distinct policy positions 0.2 and 0.7, respectively, find the number of votes for the candidate A based on voter's ideal positions.

A) 11 B) 16 C) 19 D) 23 A) 11 B) 16 C) 19 D) 23

Suppose 16 voters are distributed at different positions over the interval [0, 1], as shown below. \(\begin{array}{|l|l|l|l|lll|l|l|l|} \hline \text { Location } & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline \begin{array}{l} \text { Number of } \\ \text { voters } \end{array} & 2 & 0 & 4 & 5 & 4 & 1 & 0 & 0 & 0 \\ \hline \end{array}\) What is the median position of the voters?

A) 0.3 B) 0.4 C) 0.6 D) None of the above. A) 0.3 B) 0.4 C) 0.6 D) None of the above.

Use the following information to answer Questions Consider the following distribution of 34 voters at eight different positions over the interval [0, 1]. \[\begin{array} { | l | l | l | l | l | l | l | l | l | } \hline \text { Location } & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline \begin{array} { l } \text { Number of } \\ \text { voters } \end{array} & 2 & 3 & 5 & 7 & 7 & 5 & 3 & 2 \\ \hline \end{array}\] -This distribution of voters is best described as:

A) skewed left. B) skewed right. C) symmetric. D) bimodal. A) skewed left. B) skewed right. C) symmetric. D) bimodal.

Use the following distribution of 31 voters at seven different positions over the interval [0, 1] to answer the Questions \[\begin{array} { | l | c | c | c | c | c | c | c | } \hline \text { Position } & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.7 & 0.8 \\ \hline \begin{array} { l } \text { Number of } \\ \text { voters } \end{array} & 1 & 4 & 5 & 6 & 5 & 6 & 4 \\ \hline \end{array}\] -What is the equilibrium position in a two-candidate election?

A) 0.3 B) 0.35 C) 0.4 D) 0.47 A) 0.3 B) 0.35 C) 0.4 D) 0.47

Use the following information to answer the Questions Consider the following distribution of 32 voters at eight different positions over the interval [0, 1]. \[\begin{array} { | l | l | l | l | l | l | l | l | l | } \hline \text { Location } & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline \begin{array} { l } \text { Number of } \\ \text { voters } \end{array} & 1 & 6 & 2 & 7 & 7 & 2 & 6 & 1 \\ \hline \end{array}\] -Assume that Candidate A and Candidate B's policy positions are 0.3 and 0.8. If Candidate C enters at 0.49, what fraction of the vote does the winner receive?

A) 9/32 B) 7/16 C) 1/2 D) 9/16 A) 9/32 B) 7/16 C) 1/2 D) 9/16

Exam 12: Electing the President

Use the following information to answer the Questions Five board members of a company must decide between two other board members (A and B) for a certain position. The voters are concerned with two issues and their ideal points are represented as ordered pairs: (1, 5), (3, 4), (5, 2), (6, 4), and (6, 2). Assume that A and B announce policy positions of (1, 2) and (5, 4), respectively. -What is the equation of the line which divides the two-dimensional space separating voters preferring A over B from those preferring B over A?

(Multiple Choice)

4.8/5

(101)

Question 21

Use the following distribution of 33 voters at eight different positions over the interval [0, 1] to answer Questions Position 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of voters 1 4 6 6 6 2 5 3 -What is the median of the distribution?

(Multiple Choice)

4.9/5

(82)

Question 22

Use the following information to answer the Questions Five board members of a company must decide between two other board members (A and B) for a certain position. The voters are concerned with two issues and their ideal points are represented as ordered pairs: (1, 5), (3, 4), (5, 2), (6, 4), and (6, 2). Assume that A and B announce policy positions of (1, 2) and (5, 4), respectively. -Which candidate would win the election?

(Multiple Choice)

4.7/5

(79)

Question 23

Use the following information to answer the Questions Five board members of a company must decide between two other board members (A and B) for a certain position. The voters are concerned with two issues and their ideal points are represented as ordered pairs: (1, 5), (3, 4), (5, 2), (6, 4), and (6, 2). Assume that A and B announce policy positions of (1, 2) and (5, 4), respectively. -How many board members would vote for B based on ideal points?

(Multiple Choice)

4.8/5

(76)

Question 24

Use the following to answer the Questions A distribution of voters is uniform over [0, 1]. Candidates A and B have already entered the election and have chosen positions at 1/4 and 3/4, respectively. -If C enters at 1, who receives a plurality?

(Multiple Choice)

4.9/5

(88)

Question 25

How many votes are available in the Electoral College?

(Multiple Choice)

4.7/5

(71)

Question 26

The point on the horizontal axis of a voter distribution where half of the voters have attitudes that lie to the left and half of the voters have attitudes that lie to the right is the:

(Multiple Choice)

4.9/5

(71)

Question 27

Give an example of five data that is unimodal.

(Essay)

4.9/5

(80)

Question 28

Use the following distribution of 33 voters at eight different positions over the interval [0, 1] to answer Questions Position 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of voters 1 4 6 6 6 2 5 3 -For two candidates A and B with distinct policy positions 0.3 and 0.8, respectively, find the number of votes for the candidate A based on voter's ideal positions.

(Multiple Choice)

4.9/5

(69)

Question 29

Use the following to answer the Questions Suppose that the continuous distribution of voters' ideal points is the bent solid line below. -Assuming that Candidate A's policy position is 2.5, if Candidate B announces a policy position between 1 and 1.5, then who will be the winner of the election?

(Multiple Choice)

4.9/5

(75)

Question 30

Use the following to answer the Questions A distribution of voters is uniform over [0, 1]. Candidates A and B have already entered the election and have chosen positions at 1/4 and 3/4, respectively. -If C enters at 1/3, who receives a plurality?

(Multiple Choice)

5.0/5

(69)

Question 31

Describe the 1/3-separation obstacle.

(Essay)

4.8/5

(73)

Question 32

Use the following to answer the Questions Suppose that the continuous distribution of voters' ideal points is the bent solid line below. -Assume that Candidate A and Candidate B's policy positions are 1.5 and 2.5. If Candidate C enters at 2, who would win the election?

(Multiple Choice)

4.8/5

(83)

Question 33

Use the following distribution of 32 voters at eight different positions over the interval [0, 1] to answer the Questions Position 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 Number of voters 6 5 5 3 4 6 2 1 -For two candidates A and B with distinct policy positions 0.2 and 0.7, respectively, find the number of votes for the candidate A based on voter's ideal positions.

(Multiple Choice)

4.8/5

(80)

Question 34

Suppose 16 voters are distributed at different positions over the interval [0, 1], as shown below. Location 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of voters 2 0 4 5 4 1 0 0 0 What is the median position of the voters?

(Multiple Choice)

4.8/5

(74)

Question 35

Use the following information to answer Questions Consider the following distribution of 34 voters at eight different positions over the interval [0, 1]. Location 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of voters 2 3 5 7 7 5 3 2 -This distribution of voters is best described as:

(Multiple Choice)

4.9/5

(73)

Question 36

Use the following distribution of 31 voters at seven different positions over the interval [0, 1] to answer the Questions Position 0.1 0.2 0.3 0.4 0.5 0.7 0.8 Number of voters 1 4 5 6 5 6 4 -What is the equilibrium position in a two-candidate election?

(Multiple Choice)

4.8/5

(84)

Question 37

Use the following information to answer the Questions Seven board members of a company must decide between two other board members (A and B) for a certain position. The voters are concerned with two issues and their ideal points are represented as ordered pairs: (1, 5), (3, 3), (5, 2), (6, 4), (6, 2), (4, 4), and (3, 2). Assume that A and B announce policy positions of (2, 2) and (5, 4), respectively. -Find the midpoint of the two policy positions.

(Short Answer)

4.9/5

(76)

Question 38

Explain the justification for the Electoral College.

(Essay)

4.7/5

(93)

Question 39

Use the following information to answer the Questions Consider the following distribution of 32 voters at eight different positions over the interval [0, 1]. Location 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of voters 1 6 2 7 7 2 6 1 -Assume that Candidate A and Candidate B's policy positions are 0.3 and 0.8. If Candidate C enters at 0.49, what fraction of the vote does the winner receive?

(Multiple Choice)

4.8/5

(69)

Question 40

Use the following distribution of 33 voters at eight different positions over the interval [0, 1] to answer Questions Position 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of voters 1 4 6 6 6 2 5 3 -What is the median of the distribution?

Use the following to answer the Questions A distribution of voters is uniform over [0, 1]. Candidates A and B have already entered the election and have chosen positions at 1/4 and 3/4, respectively. -If C enters at 1, who receives a plurality?

How many votes are available in the Electoral College?

The point on the horizontal axis of a voter distribution where half of the voters have attitudes that lie to the left and half of the voters have attitudes that lie to the right is the:

Give an example of five data that is unimodal.

Use the following to answer the Questions Suppose that the continuous distribution of voters' ideal points is the bent solid line below. -Assuming that Candidate A's policy position is 2.5, if Candidate B announces a policy position between 1 and 1.5, then who will be the winner of the election?

Use the following to answer the Questions A distribution of voters is uniform over [0, 1]. Candidates A and B have already entered the election and have chosen positions at 1/4 and 3/4, respectively. -If C enters at 1/3, who receives a plurality?

Describe the 1/3-separation obstacle.

Use the following to answer the Questions Suppose that the continuous distribution of voters' ideal points is the bent solid line below. -Assume that Candidate A and Candidate B's policy positions are 1.5 and 2.5. If Candidate C enters at 2, who would win the election?

Suppose 16 voters are distributed at different positions over the interval [0, 1], as shown below. Location 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of voters 2 0 4 5 4 1 0 0 0 What is the median position of the voters?

Use the following information to answer Questions Consider the following distribution of 34 voters at eight different positions over the interval [0, 1]. Location 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of voters 2 3 5 7 7 5 3 2 -This distribution of voters is best described as:

Use the following distribution of 31 voters at seven different positions over the interval [0, 1] to answer the Questions Position 0.1 0.2 0.3 0.4 0.5 0.7 0.8 Number of voters 1 4 5 6 5 6 4 -What is the equilibrium position in a two-candidate election?

Explain the justification for the Electoral College.

Urban Services

Business Efficiency

Planning and Scheduling

Linear Programming

Exploring Data: Distributions

Exploring Data: Relationships

Data for Decisions

Probability: the Mathematics of Chance

Social Choice: the Impossible Dream

The Manipulability of Voting Systems

Weighted Voting Systems

Fair Division

Apportionment

Game Theory: the Mathematics of Competition

Identification Numbers

Information Science

Growth and Form

Symmetry and Patterns

Tilings

Savings Models

Borrowing Models

The Economics of Resources

Filters