MGF 1107
SUPPLEMENTAL HOMEWORK/ CHAPTER 14
1. Memorial Hospital has a nursing staff of 160 nurses working in four shifts: A (7am-1pm), B (1pm-7pm), C (7pm-1am), and D (1am-7am). The number of nurses apportioned to each shift is based on the average number of patients per shift, given in the following table:
Shift |
Average number of patients |
|
A | 662 |
|
B | 534 |
|
C | 266 |
|
D | 138 |
a) Find the standard divisor. Explain what the standard divisor represents in this problem.
b) Apportion the nurses among the shifts using Adams’ method.
2. The Dyersville Area Rapid Transit Service (DARTS) operates 6 bus routes (A, B, C, D, E, and F) and 130 buses. The buses are apportioned among the routes based on the average number of daily passengers per route, given in the following table:
Route |
Average number of passengers |
|
A | 45,300 |
|
B | 31,070 |
|
C | 20,490 |
|
D | 14,160 |
|
E | 10,260 |
|
F | 8,720 |
a) Find the standard divisor. Explain what the standard divisor represents in this problem.
b) Apportion the buses among the routes using Adams’ method.
3. Everglades University is made up of five different schools: Agriculture, Business, Education, Humanities, and Science. The 119 faculty positions at EU are apportioned to the various schools based on the schools’ respective enrollments, given in the following table:
School |
Enrollment |
|
Agriculture | 4050 |
|
Business | 2970 |
|
Education | 2365 |
|
Humanities | 1460 |
|
Science | 1055 |
a) Find the standard divisor. Explain what the standard divisor represents in this problem.
b) Apportion the faculty positions among the schools using Adams’ method.
4. Twenty-eight seats shall be apportioned to five undergraduate colleges and schools of the Johnson City campus of East Tennessee State University in the following manner. The Secretary of Interior, with the assistance of the advisor for the Student Government Association, will obtain the following total student enrollment figures from the Fall Census:
A. College of Arts and Science;
B. College of Business and Technology;
C. College of Education;
D. College of Nursing;
E. College of Public and Allied Health;
Total of the five above represents the total Johnson City campus undergraduate student enrollment figure. The Secretary of Interior, assisted by the advisor for the Student Government Association, will determine apportionment of the colleges and schools by:
A. Dividing the student enrollment figure of each college or school by the total Johnson City campus undergraduate student enrollment figure to get a percentage.
B. The percentage obtained in A) above will be multiplied times the twenty-eight seats apportioned to respective college or school for that academic year’s general election. In no case will apportionment to any college or school be less than one seat.
What is wrong with this method of apportionment?
5. The bylaws
of the Georgia Tech Undergraduate Student Government read as follows: Each
degree granting school or college shall elect representatives based on the fall
semester enrollment figures prior to the election according to the following formula:
number of enrolled students divided by 400 equals the number of representatives.
The number of representatives shall be rounded up if a decimal is equal to or
greater than 5.
Source: http://cyberbuzz.gatech.edu/sga/undergraduate/legis/bylaws.html
In class, we saw that using this standard rounding method often doesn’t work (without the modifications of Webster’s method). Why does it work for Georgia Tech?
6. A tiny country with population 4693 consists of 6 states with populations as follows:
| A |
B |
C |
D |
E |
F |
|
1518 | 1305 | 865 | 499 | 296 | 210 |
Use the Hill-Huntington method to apportion the 23 seats in the legislature.
7. Of the apportionment methods we studied, which are not divisor methods?
8. Of the apportionment methods we studied, which are susceptible to the Alabama paradox?
9. Of the apportionment methods we studied, which satisfy the quota condition?
10. Of the apportionment methods we studied, which avoid the population paradox and satisfy the quota condition?
11. Miami-Dade county is forming an all-star Khoury League baseball team consisting of 26 players to play in the state tournament. The county supervising committee has decided to apportion athletes to the team from each local Khoury league according to the number of players in the league.
| League |
Tamiami |
Continental |
Howard-Palmetto |
West Kendall |
Miami Lakes |
Flagami |
|
# of players | 1002 | 80 | 664 | 106 | 92 | 116 |
Determine the composition of the all-star team using:
a) Hamilton’s method
b) Jefferson’s method
c) Adams’ method
d) Webster’s method
e) Hill-Huntington method
f) Do any of the methods result in a quota violation?
12. Of the five methods we studied, which have never been used to apportion the U.S. House of Representatives?
13. Of the four divisor methods we studied, which are unbiased?
14. Use the following figures from 1970.
| |
Quota |
Adams apportionment |
Hill-Huntington apportionment |
Webster apportionment |
Jefferson apportionment |
|
Total of 16 Largest states | 300.7935 | 291 | 300 | 300 | 309 |
|
Total of 16 smallest states | 25.4493 | 33 | 27 | 25 | 22 |
Compute the bias percentage (rounded off to one decimal place) in favor of small states for
a) Adams’ method
b) Hill-Huntington method
c) Webster’s method
d) Jefferson’s method
15. Of the four divisor methods we studied, which one is always biased in favor of
a) small states?
b)
large states?
16. The 2000 apportionment, which was done with the Hill-Huntington
method, apportioned 53 seats to California and 32 seats to Texas. California's
population was 33,930,798 and Texas' was 20,903,994.
a) Find California's
representative share. Express your answer in microseats/person, rounded to two
decimal places.
b) Find Texas' representative share. Express your answer in
microseats/person, rounded to two decimal places.
c) Which state is favored
in this apportionment?
d) Find the absolute difference in representative shares,
rounded to two decimal places.
e) Find the relative difference in representative
shares, rounded to two decimal places.
f) If we move a seat from California
to Texas, would this make the relative difference in respresentative shares smaller?
17.
A country has four states, A, B, C, and D. Its house of representatives has 100
members, and by law, the Hamilton method is used to apportion it. A new census
is taken, and the house is reapportioned. Here are the data:
State | Old
census | New Census |
A | 5,525,381 | 5,657,564 |
B | 3,470,152 | 3,507,464 |
C | 3,864,226 | 3,885,693 |
D |
201,203 | 201,049 |
Totals | 13,060,962 | 13,251,770 |
a) Apportion the house using the old census.
b) Reapportion,
using the new census.
c) Explain how this is an example of the population paradox.
Province |
Population |
|
Matanzas |
43,000 |
|
Cienfuegos |
32,800 |
|
Camagüey |
24,200 |
Answers:
1a) 10. The average number of patients per nurse per day.
1b) A: 66, B: 53, C: 27, D: 14
2a) 1000. The average number of passengers per bus per day.
2b) A: 45, B: 31, C: 20, D: 14, E: 11, F: 9
3a) 100. The average number of students per faculty member.
3b) Agriculture: 40, Business: 29, Education: 24, Humanities 15, Science: 11
4) It doesn’t specify how to handle the results to step B when they are not whole numbers.
5) Georgia Tech doesn’t specify the total size of the legislative body.
6) A: 7, B: 6, C: 4, D: 3, E: 2, F: 1
7) Only Hamilton’s method.
8) Only Hamilton’s method.
9) Only Hamilton’s method.
10) None of them.
11)
| |
Tamiami |
Continental |
Howard-Palmetto |
West Kendall |
Miami Lakes |
Flagami |
|
a) | 13 | 1 | 8 | 1 | 1 | 2 |
|
b) | 13 | 1 | 9 | 1 | 1 | 1 |
|
c) | 11 | 1 | 8 | 2 | 2 | 2 |
|
d) | 13 | 1 | 9 | 1 | 1 | 1 |
|
e) | 13 | 1 | 8 | 1 | 1 | 2 |
f) Yes, Adams’ method.
12. Only Adams’ method.
13.
Only Webster’s method.
| 14a) 32.9% |
14b) 6.4% | 14c) –1.5% | 14d) –16.3% |
| 15a) Adams’ method | 15b) Jefferson’s method |
| 16a) 1.56 microseats/person | 16b) 1.53 microseats/person |
16c) California | 16d) 0.03 microseats/person | 16e) 1.96% |
16f) No, the Hill-Huntington method always produces the apportionment
that minimizes the relative difference in representative shares.
17a) A gets 42 members, B gets 27, C gets 30, and D gets 1.
17b) A gets 43 members, B gets 26, C gets 29, and D gets 2.
17c) States B and C had increased populations and decreased apportionments.
Although the population of state D decreased slightly, its apportionment increased.
18.
|
Matanzas |
5 |
|
Cienfuegos |
4 |
|
Camagüey |
2 |