Apportionment, Weighted Voting, and Normal Distribution Review

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

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).

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

They are always rounded down (floored).

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

Traditional (ordinary) rounding to the nearest integer.

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.

What is the standard divisor in Hamilton’s method?

Total population divided by the total number of seats.

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

Jefferson; its round-down rule benefits larger states.

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

The population paradox.

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

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

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.

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

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

In a weighted voting system, what is the quota?

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

How is a weighted voting system typically written?

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

When is a coalition winning in a weighted voting system?

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

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.

What is a veto power voter?

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

Define a dummy voter.

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

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.

How many possible coalitions exist for n voters?

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

How many coalitions are there with four voters?

2⁴ = 16 coalitions.

What is a critical voter in a coalition?

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

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.

What does a BPI value measure?

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

What is the total area under any normal distribution curve?

1 (or 100% of the probability).

Describe the symmetry of a normal distribution.

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

Is the normal curve asymptotic to the horizontal axis?

Yes; it approaches the axis but never touches it.

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.

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

About 68.3% (roughly 68%).

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

Mean μ = 0 and standard deviation σ = 1.

What does a z-score represent?

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

What does a z-score of 0 indicate?

The data point equals the population mean.

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

0.5 (half of the area).

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

z = (x − μ) ⁄ σ.

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

NORM.S.DIST(z, TRUE).

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).

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.

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.

Which apportionment method violates the quota criterion most frequently?

Jefferson (because it always rounds down).