Practice TEST: Chapter 14

Transcription

1 TOPICS Practice TEST: Chapter 14 Name: Period: Date: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the given information to answer the question. 1) The preference ballots for presidency of the Garage Punk Appreciation Club (A, B, and C) 1) are shown. Fill in the number of votes in the first row of the preference table. ACB ACB ACB CBA BAC CBA CBA BAC BAC ACB CBA ACB ACB BAC BAC ACB Number of Votes First choice A B C Second choice C A B Third choice B C A Use the preference table to answer the question. 2) Four students are running for president of their dormitory: Debra (D), Farah (F), Jorge (J), 2) and Hillary (H). The votes of their fellow students are summarized in the following preference table. Number of Votes First choice D F J F H Second choice F J F J J Third choice H H H D D Fourth choice J D D H F Who is declared the new president using the plurality method? 3) Diners at the Joie de Vivre restaurant answer a questionnaire about their favorite course in 3) a French meal. The choices are: Appetizer (A), Entree (E), and Dessert (D). Their votes are summarized in the following table. Number of Votes First choice E D A D Second choice A A D E Third choice D E E A Which course is selected as the most favorite using the Borda count method? 1

2 4) Four students are running for president of their graduating class: Debra (D), Farah (F), 4) Jorge (J), and Hillary (H). The votes of their fellow students are summarized in the following preference table. Number of Votes First choice J F J F H Second choice D J F J J Third choice F H H D D Fourth choice H D D H F Who is declared the new president using the plurality-with-elimination method? 5) Diners at the Joie de Vivre restaurant answer a questionnaire about their favorite course in 5) a French meal. The choices are: Appetizer (A), Entree (E), and Dessert (D). Their votes are summarized in the following table. Number of Votes First choice E D A D Second choice A A D E Third choice D E E A Which course is selected as the most favorite using the pairwise comparison method? 6) The preference table shows the results of an election among three candidates, A, B, and C. 6) Number of votes First choice A B B Second choice B C A Third choice C A C (a) Using the plurality method, who is the winner? (b) Is the majority criterion satisfied? 7) The preference table shows the results of an election among three candidates, A, B, and C. 7) Number of votes First choice A B B Second choice B C A Third choice C A C (a) Using the plurality method, who is the winner? (b) Is the head-to head criterion satisfied? 2

3 8) The preference table shows the results of a straw vote among three candidates, A, B, and C. 8) Number of votes First choice A B C Second choice B C B Third choice C A A (a) Using the plurality-with-elimination method, which candidate wins the straw vote? (b) In the actual election, the 3 voters in the last column who voted C, B, and A, in that order, change their votes to A, B, C. Using plurality-with-elimination method, which candidate wins the actual election. (c) Is the monotonicity criterion satisfied? 9) The preference table shows the results of an election among three candidates, A, B, and C. 9) Number of votes First choice A B C Second choice B C B Third choice C A A (a) Using the plurality method, who is the winner? (b) The voters in the two columns on the right move their last-place candidates from last place to first place. Construct a new preference table for the election. Using the table and the plurality method, who is the winner? (c) Suppose that candidate C drops out of the new table, but the winner is still chosen by the plurality method. Is the irrelevant alternatives criterion satisfied? 10) Choose the sentence or sentences that accurately restate Arrow's Impossibility Theorem.(more than 10) one may apply) I. It is mathematically impossible for any democratic voting system to satisfy any of the four fairness criteria. II. It is mathematically impossible for any democratic voting system to satisfy all of the four fairness criteria. III. It is mathematically impossible for any democratic voting system to satisfy some of the four fairness criteria. IV. It is mathematically impossible for any democratic voting system to satisfy any more than one of the four fairness criteria. A) Statement I is true. B) Statement II is true. C) Statement III is true. D) Statement IV is true. 3

4 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve. 11) A country is made up of four regions A, B, C, and D. The population of each region, in 11) thousands, is given in the following table. Region A B C D Total Population (in thousands) According to the country's constitution, the congress will have 40 seats, divided among the four regions according to their respective populations. (a) Find the standard divisor, in thousands. (b) How many people are there for each seat in the congress? Use the table to answer the question. 12) For the apportionment of 50 seats in congress among four states, the standard quotas are as 12) shown in the following table. Region A B C D Standard quota Create an extended table by entering the final quotas using Hamilton's method. Standard quota Quota using Hamilton's method Region A B C D Solve the problem. 13) For the apportionment of 60 seats among four states, the standard quotas are as shown in 13) the following table. State A B C D Standard quota Create an extended table by showing each state's upper quota and lower quota. State A B C D Standard quota Upper quota Lower quota 4

5 14) An organization helping to provide meals to city shelters for the homeless has a 14) membership of 60 volunteers. They are assigned among the four city areas A, B, C, and D in proportion to the number of people fed in the respective areas. The numbers of people fed at the city shelters in each city area are shown in the following table. Use Jefferson's method to extend the table and apportion the 60 volunteers among the city areas. Number of volunteers 15) An organization helping to provide meals to city shelters for the homeless has a 15) membership of 90 volunteers. They are assigned among the four city areas A, B, C, and D in proportion to the number of people fed in the respective areas. The numbers of people fed at the city shelters in each city area are shown in the following table. Use Adams's method to extend the table and apportion the 90 volunteers among the city areas. Number of volunteers 16) An organization helping to provide meals to city shelters for the homeless has a 16) membership of 60 volunteers. They are assigned among the four city areas A, B, C, and D in proportion to the number of people fed in the respective areas. The numbers of people fed at the city shelters in each city area are shown in the following table. Use Webster's method to extend the table and apportion the 60 volunteers among the city areas. Number of volunteers 5

6 17) In the year 2001, the economics department of a university had 25 teaching assistants (TAs) to be 17) divided among three courses, according to their respective enrollments. Table 1 shows the courses and the standard quotas as calculated for 25 TAs and 26 TAs Table 2 shows the same data for year 2002, when the enrollment levels were different. TABLE 1 (2001) Course General Theory Business Standard quota with 25 TAs Standard quota with 26TAs TABLE 2 (2002) Course General Theory Business Standard quota with 25 TAs Standard quota with 26TAs The Alabama paradox is said to exist when an increase in the total number of items to be apportioned results in the loss of an item for a group. Using the Hamilton method to determine final quotas, decide which of the statements below is true. A) The Alabama paradox would have occurred in B) The Alabama paradox would have occurred in C) The Alabama paradox would have occurred in both 2001 and D) The Alabama paradox would not have occurred. 18) Four regions (A, B, C, and D) in a country control seats in a national congress apportioned 18) according to the respective populations in each region. The table shows the percentage of population growth in each region from 1991 to Population Growth 1991 to 1992 Region A 2.3% Region B 1.3% Region C 2.4% Region D 2.9% The following two statements were made about the information contained in the table: I. The population paradox occurred if Region B lost a seat to region A in II. The population paradox occurred if Region C lost a seat to region D in Which, if either, of these statements is accurate? (list all that apply) A) Statement I is true. B) Statement II is true. 6

7 19) New-states Paradox: 19) The addition of a new group changes the apportionments of other groups. Suppose there are 40 states in a country where congressional seats are alloted to the states in proportion to the population of each respective state. Total population is 12 million people. Suppose also that a 41st state is admitted, adding its own population to that of the nation. If the new-states paradox occurs, which of the following statements are true: (list all that apply) I. In the original 40 states, only one state will end up with a different number of congressional seats. II. In the original 40 states, at least two states will end up with a different number of congressional seats. III. The original 40 states will all retain the same number of seats if the 41st state has a population of under 500,000. IV. If the 41st state has a population greater than one million, at least two of the original states will end up with a different number of congressional seats. A) Statement I is true. B) Statement II is true. C) Statement III is true. D) Statement IV is true. 7