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 0.55, 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

Use the following information to answer the Questions Suppose 36 voters are distributed at nine different positions over the interval [0, 1], as suggested by this incomplete table. \[\begin{array} { | l | l | l | l | l | l | l | l | l | l | } \hline \text { Location } & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline \begin{array} { l } \text { Number } \\ \text { of voters } \end{array} & ? & ? & ? & ? & ? & ? & ? & ? & ? \\ \hline \end{array}\] -Which situation below would result in a skewed distribution?

A) 18 voters at 0.3 and 18 voters at 0.7 B) three voters at each of the nine positions C) all 36 voters at 0.1 D) nine voters at each of these four positions: 0.2, 0.4, 0.6, 0.8 A) 18 voters at 0.3 and 18 voters at 0.7 B) three voters at each of the nine positions C) all 36 voters at 0.1 D) nine voters at each of these four positions: 0.2, 0.4, 0.6, 0.8

Use the following distribution of 32 voters at eight different positions over the interval [0, 1] to answer the Questions \[\begin{array} { | l | c | c | c | c | c | c | c | c | } \hline \text { Position } & 0.1 & 0.2 & 0.3 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline \begin{array} { l } \text { Number of } \\ \text { voters } \end{array} & 6 & 5 & 5 & 3 & 4 & 6 & 2 & 1 \\ \hline \end{array}\] -What is the equilibrium position in a two-candidate election?

A) 0.3 B) 0.4 C) 0.5 D) 0.6 A) 0.3 B) 0.4 C) 0.5 D) 0.6

Consider the following distribution of 35 voters at eight different positions over the interval [0, 1]. \[\begin{array} { | l l | l | l l l | l | l l | } \hline \text { Location } & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \text { Number of } & 4 & 7 & 4 & 2 & 3 & 7 & 4 & 4 \\ \text { voters } & & & & & & & & \end{array}\] 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 information to answer the Questions Consider the following distribution of 34 voters at eight different positions over the interval [0, 1]. \[\begin{array} { | l | l | l | l | l | l | l | l | l | } \hline \text { Location } & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline \begin{array} { l } \text { Number of } \\ \text { voters } \end{array} & 3 & 6 & 7 & 4 & 3 & 4 & 4 & 3 \\ \hline \end{array}\] -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.

Exam 12: Electing the President

Use the following information to answer the Questions Suppose that the continuous distribution of voters' ideal points is the bent solid line below. -Explain why the bent solid line describes a continuous distribution.

(Essay)

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Question 61

Give an example of a voter distribution over the interval [0,1] that is skewed to the left.

(Essay)

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Question 62

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 midpoint of the two policy positions?

(Multiple Choice)

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Question 63

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, what fraction of the vote does the winner receive?

(Multiple Choice)

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Question 64

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. -Which candidate would be elected for the position based on the ideal points?

(Short Answer)

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Question 65

What is the largest value of the extended median if the voters' ideal points are 2, 2, 4, 4, 6, 8, 8, 8, 10, 10?

(Multiple Choice)

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Question 66

What is $\sum _ { j = 1 } ^ { 4 } 2 ?$

(Multiple Choice)

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Question 67

A candidate selects a point on the horizontal axis of a voter distribution where no position can guarantee a better outcome, no matter what position another candidate adopts. This position is the:

(Multiple Choice)

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Question 68

Explain the extended median of a discrete distribution.

(Essay)

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Question 69

Which statement is correct for the bent solid line given below?

(Multiple Choice)

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Question 70

Use the following information to answer the Questions Suppose that 17 voters have ideal points as given in the following table. Ideal points 0 1 2 3 4 5 6 Number of voters 1 2 3 5 3 2 1 Candidate A announces a policy position of 1 and Candidate B announces a policy position of 5. -Suppose that a voter is added with ideal points of 3. Is there a 2/3-separation opportunity for a third Candidate C to enter the race?

(Essay)

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Question 71

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 0.55, what fraction of the vote does the winner receive?

(Multiple Choice)

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Question 72

Use the following information to answer the Questions Suppose 36 voters are distributed at nine different positions over the interval [0, 1], as suggested by this incomplete table. Location 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of voters ? ? ? ? ? ? ? ? ? -Which situation below would result in a skewed distribution?

(Multiple Choice)

4.8/5

(80)

Question 73

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 -What is the equilibrium position in a two-candidate election?

(Multiple Choice)

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Question 74

Suppose voters are distributed so that they are only located at certain positions along the left-right continuum. What is this type of distribution called?

(Short Answer)

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Question 75

Consider the following distribution of 35 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 4 7 4 2 3 7 4 4 voters This distribution of voters is best described as:

(Multiple Choice)

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Question 76

How many electoral votes are needed to win an election?

(Multiple Choice)

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(85)

Question 77

Use the following information to answer the Questions Suppose that the continuous distribution of voters' ideal points is the bent solid line below. -Assuming Candidate A and Candidate B's policy positions are 2.7 and 3.3, determine whether Candidate C can enter and win the election.

(Essay)

4.8/5

(78)

Question 78

Use the following information to answer the 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 3 6 7 4 3 4 4 3 -This distribution of voters is best described as:

(Multiple Choice)

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(74)

Question 79

Find the range of values that make up the extended median if the voters' ideal points are 2, 3, 4, 4, 6, 8, 8, 8, 9, 10.

(Essay)

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(90)

Question 80

Use the following information to answer the Questions Suppose that the continuous distribution of voters' ideal points is the bent solid line below. -Explain why the bent solid line describes a continuous distribution.

Give an example of a voter distribution over the interval [0,1] that is skewed to the left.

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, what fraction of the vote does the winner receive?

What is the largest value of the extended median if the voters' ideal points are 2, 2, 4, 4, 6, 8, 8, 8, 10, 10?

What is $\sum _ { j = 1 } ^ { 4 } 2 ?$

A candidate selects a point on the horizontal axis of a voter distribution where no position can guarantee a better outcome, no matter what position another candidate adopts. This position is the:

Explain the extended median of a discrete distribution.

Which statement is correct for the bent solid line given below?

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 -What is the equilibrium position in a two-candidate election?

Suppose voters are distributed so that they are only located at certain positions along the left-right continuum. What is this type of distribution called?

Consider the following distribution of 35 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 4 7 4 2 3 7 4 4 voters This distribution of voters is best described as:

How many electoral votes are needed to win an election?

Use the following information to answer the Questions Suppose that the continuous distribution of voters' ideal points is the bent solid line below. -Assuming Candidate A and Candidate B's policy positions are 2.7 and 3.3, determine whether Candidate C can enter and win the election.

Use the following information to answer the 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 3 6 7 4 3 4 4 3 -This distribution of voters is best described as:

Find the range of values that make up the extended median if the voters' ideal points are 2, 3, 4, 4, 6, 8, 8, 8, 9, 10.

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

Exam 12: Electing the President

Use the following information to answer the Questions Suppose that the continuous distribution of voters' ideal points is the bent solid line below. -Explain why the bent solid line describes a continuous distribution.

Give an example of a voter distribution over the interval [0,1] that is skewed to the left.

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, what fraction of the vote does the winner receive?

What is the largest value of the extended median if the voters' ideal points are 2, 2, 4, 4, 6, 8, 8, 8, 10, 10?

What is ∑j=142?\sum _ { j = 1 } ^ { 4 } 2 ?∑j=14​2?

A candidate selects a point on the horizontal axis of a voter distribution where no position can guarantee a better outcome, no matter what position another candidate adopts. This position is the:

Explain the extended median of a discrete distribution.

Which statement is correct for the bent solid line given below?

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 -What is the equilibrium position in a two-candidate election?

Suppose voters are distributed so that they are only located at certain positions along the left-right continuum. What is this type of distribution called?

Consider the following distribution of 35 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 4 7 4 2 3 7 4 4 voters This distribution of voters is best described as:

How many electoral votes are needed to win an election?

Use the following information to answer the Questions Suppose that the continuous distribution of voters' ideal points is the bent solid line below. -Assuming Candidate A and Candidate B's policy positions are 2.7 and 3.3, determine whether Candidate C can enter and win the election.

Use the following information to answer the 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 3 6 7 4 3 4 4 3 -This distribution of voters is best described as:

Find the range of values that make up the extended median if the voters' ideal points are 2, 3, 4, 4, 6, 8, 8, 8, 9, 10.

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

What is $\sum _ { j = 1 } ^ { 4 } 2 ?$