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Chapter 2: Problem 66

Political scientists have discovered the following empirical rule, known as the "cube rule," which gives the relationship between the proportion of seats in the House of Representatives won by Democratic candidates \(s(x)\) and the proportion of popular votes \(x\) received by the Democratic presidential candidate: $$ s(x)=\frac{x^{3}}{x^{3}+(1-x)^{3}} \quad(0 \leq x \leq 1) $$ Compute \(s(0.6)\) and interpret your result.

Short Answer

Expert verified
The short answer based on the step-by-step solution is: When the Democratic presidential candidate receives 60% of the popular vote, the Democratic candidates would win approximately 77.14% of the seats in the House of Representatives, as calculated using the cube rule, \(s(0.6) \approx 0.7714\).

Step by step solution

01

Plug the given value of x into the cube rule

The cube rule is given as \(s(x) = \frac{x^3}{x^3 + (1-x)^3}\), and we need to find \(s(0.6)\). So, let's plug \(x = 0.6\) into the equation: $$ s(0.6) =\frac{0.6^3}{0.6^3 + (1-0.6)^3} $$
02

Compute the numerator and denominator of the fraction

We will first compute \(0.6^3\) and \((1-0.6)^3\): $$ \space 0.6^3 = (0.6)(0.6)(0.6) = 0.216\\ (1-0.6)^3 = (0.4)(0.4)(0.4) = 0.064 $$ Now, compute the sum in the denominator as follows: $$ 0.216 + 0.064 = 0.280 $$
03

Calculate s(0.6)

Now, we'll divide the numerator by the computed sum of the denominator: $$ s(0.6) = \frac{0.216}{0.280} \approx 0.7714 $$
04

Interpret the result

The calculated value of \(s(0.6)\) is approximately \(0.7714\). This means that if the Democratic presidential candidate receives 60% of the popular vote, the Democratic candidates would win roughly 77.14% of the seats in the House of Representatives.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Empirical Rule
The Empirical Rule, often associated with statistics, is a guideline that applies to normal distributions. In essence, it's a shorthand way to describe how data points spread around a mean value. In a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% within three standard deviations.

While the term 'empirical rule' is broadly used in statistics, it is also applied within political science in a specific context, such as the 'cube rule' in the given exercise. Adapted to political science, it represents a pattern or formula, derived from observation of historical data, that tries to predict an outcome based on certain inputs鈥攊n this case, the relationship between popular votes and seats won.
Cube Rule Politics
Cube Rule Politics refers to a mathematical model or rule that aims to represent the nonlinear relationship between the proportion of votes a political party receives and the proportion of seats they gain as a result. In political science, the cube rule is one such empirical rule. The exercise presented here uses the cube rule to depict how the number of seats in the House of Representatives that Democratic candidates would win scales with the proportion of popular votes they receive.

The formula given, \( s(x) = \frac{x^3}{x^3 + (1-x)^3} \), is striking in its use of the cube of the vote share, suggesting that small changes in popular vote share can lead to disproportionately large changes in seat share. This demonstrates the cube rule's prediction of a leveraged effect of the popular vote on seats won, which is a feature of political systems with single-member districts and plurality voting.
Proportion Calculations
Proportion calculations are a fundamental part of applied mathematics. They are used to determine how parts of a whole relate to each other. This is especially useful in fields like chemistry, architecture, and economics. In the context of the 'cube rule,' we are looking at a specific proportion calculation that reflects the distribution of seats in a legislative body based on voting percentages.

When calculating proportions, one aims to find a fraction or percentage that expresses the relationship between a part and the whole. The calculation done in the exercise example, \(s(0.6) = \frac{0.216}{0.280} \approx 0.7714\), showcases this process. It uses the cube rule to translate the percentage of popular votes received into the expected percentage of seats won. Understanding proportion calculations is essential for interpreting models like the cube rule and for grasping more complex relationships between quantitative entities in various disciplines.

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