Apportionment, Weighted Voting, and Normal Distribution Review

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These flashcards review major concepts on apportionment methods, weighted voting, and normal‐distribution statistics likely to appear on the upcoming exam.

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37 Terms

1
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Does the Hamilton (largest-remainder) method always satisfy the quota criterion?

No. Although it begins with lower quotas, Hamilton can violate quota (e.g., the Alabama paradox).

2
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In Jefferson’s method, how are standard quotas rounded?

They are always rounded down (floored).

3
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What rounding rule does Webster’s method apply to the quotas?

Traditional (ordinary) rounding to the nearest integer.

4
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Is the Huntington–Hill method immune to all apportionment paradoxes?

No. It avoids several paradoxes but, by the Balinski–Young Theorem, no method can avoid them all while meeting quota.

5
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What is the standard divisor in Hamilton’s method?

Total population divided by the total number of seats.

6
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Which apportionment method tends to favor larger states—Jefferson or Hamilton?

Jefferson; its round-down rule benefits larger states.

7
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Which paradox occurs when a group’s population grows yet its seat share drops?

The population paradox.

8
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What does the Balinski–Young Theorem assert about apportionment methods?

No method can simultaneously satisfy the quota criterion and avoid all apportionment paradoxes.

9
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How does the Huntington–Hill method decide whether to round a quota up or down?

Compare the quota to the geometric mean √[A(A+1)]; round up if it exceeds that value, otherwise round down.

10
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Which apportionment methods adjust a modified divisor by trial-and-error until the seat total is correct?

Jefferson, Webster, and Huntington–Hill (divisor methods).

11
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In a weighted voting system, what is the quota?

The minimum total weight required to pass a motion, not the number of players.

12
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How is a weighted voting system typically written?

W = q : w₁, w₂, …, wₙ (quota followed by the individual weights).

13
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When is a coalition winning in a weighted voting system?

When the sum of its weights is greater than or equal to the quota.

14
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What is a dictator in a weighted voting system?

A single voter whose weight alone meets or exceeds the quota, letting them pass motions unilaterally.

15
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What is a veto power voter?

A voter whose participation is necessary in every winning coalition but who cannot pass a motion alone.

16
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Define a dummy voter.

A voter who is never critical; their presence or absence never changes the outcome.

17
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What is a blocking (or stopping) coalition?

A group of voters that can prevent a measure from passing, typically because the remaining voters cannot reach the quota without them.

18
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How many possible coalitions exist for n voters?

2ⁿ coalitions (including the empty set and the grand coalition).

19
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How many coalitions are there with four voters?

2⁴ = 16 coalitions.

20
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What is a critical voter in a coalition?

One whose removal would change the coalition from winning to losing.

21
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How is a voter’s Banzhaf Power Index (BPI) computed?

Count the number of times the voter is critical in winning coalitions and divide by the total number of critical occurrences for all voters.

22
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What does a BPI value measure?

The relative influence or power of a voter in a weighted voting system.

23
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What is the total area under any normal distribution curve?

1 (or 100% of the probability).

24
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Describe the symmetry of a normal distribution.

It is perfectly symmetric; the mean, median, and mode coincide at the center.

25
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Is the normal curve asymptotic to the horizontal axis?

Yes; it approaches the axis but never touches it.

26
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State the empirical (68-95-99.7) rule.

Approximately 68% of data lie within 1 σ, 95% within 2 σ, and 99.7% within 3 σ of the mean in a normal distribution.

27
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What proportion of data lie within one standard deviation of the mean in a normal distribution?

About 68.3% (roughly 68%).

28
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What are the mean and standard deviation of the standard normal distribution?

Mean μ = 0 and standard deviation σ = 1.

29
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What does a z-score represent?

The number of standard deviations a data value is above (>0) or below (<0) the mean.

30
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What does a z-score of 0 indicate?

The data point equals the population mean.

31
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What is the probability that z < 0 in a standard normal distribution?

0.5 (half of the area).

32
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How do you convert a raw score x to a z-score?

z = (x − μ) ⁄ σ.

33
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Which Excel function returns cumulative probabilities for the standard normal distribution?

NORM.S.DIST(z, TRUE).

34
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According to the Central Limit Theorem, what happens to the sampling distribution of the mean as sample size increases?

It approaches a normal distribution, regardless of the population’s shape (especially when n ≥ 30).

35
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Do all data sets follow a normal distribution?

No; normality is common but not universal, so normal-ity should be checked before applying normal-based methods.

36
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In a normal distribution, are extreme values more or less frequent than central values?

Less frequent; the bell-shaped curve’s tails taper, making extremes rarer than values near the mean.

37
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Which apportionment method violates the quota criterion most frequently?

Jefferson (because it always rounds down).