Question

A poll of 400 randomly selected likely voters in Seanoa City was taken to determine the support for the mayoral candidates in the upcoming election. Of the likely voters selected, 190 stated that they are likely to vote for Candidate \(\mathrm{A}\). If the conclusion is drawn that "approximately 3,120 voters are likely to vote for Candidate A," which of the following is closest to the number of likely voters in Seanoa City? A) 1,482 B) 3,120 C) 4,741 D) 6,568 $$ \begin{gathered} y^2=21-x \\ x=5 \end{gathered} $$

Short answer

The closest number of total likely voters in Seanoa City is approximately 6,568.

Step-by-step solution

Set up the proportion.

We can set up the proportion as follows: \(\dfrac{\text{Number of voters who support A in the sample}}{\text{Total number of voters in the sample}} = \dfrac{\text{Estimated number of voters who support A in the entire city}}{\text{Total number of voters in the entire city}}\)

Plug in the given information.

We are given that there are 190 voters who support Candidate A in the sample of 400 voters, and an estimated 3,120 voters who support Candidate A in the entire city. Let x represent the total number of voters in the entire city. We can now plug in these values into our proportion: \(\dfrac{190}{400}=\dfrac{3,120}{x}\)

Cross-multiply.

To solve for x, we need to cross-multiply: \(190x = 3,120 * 400\)

Solve for x.

Divide both sides of the equation by 190: \(x = \dfrac{3,120 * 400}{190}\) Calculate the result: \(x ≈ 6,568\)

Answer the question.

The closest number of total likely voters in Seanoa City is approximately 6,568. Thus, the correct answer is: D) 6,568

Key concepts

Polling Methodology

Polling methodology is the process used to gather data about a particular group of people. In the context of elections, polls are conducted to estimate how many people might vote for a particular candidate. Pollsters often randomly select a sample of individuals from the larger population to represent everyone's opinions.

• This random selection helps ensure that the sample is unbiased and reflects the general preferences of the entire population.
• The sample size, such as the 400 likely voters in our problem, is critical because it must be large enough to provide a reliable estimate but manageable to analyze.

Once data is collected, it is analyzed to make predictions about the larger population, such as estimating support for a mayoral candidate.
The goal of a good polling methodology is to minimize errors and inaccuracies in predicting electoral outcomes.

Mathematical Modeling

Mathematical modeling involves using mathematical constructs to represent real-world scenarios. In our polling problem, we model the relationship between the voters sampled and the entire voter population using a proportion.

• This model helps us predict the overall support for a candidate based on the sample data.
• Using mathematical models in polling allows for precise calculations, facilitating better predictions.

Such models are especially useful in determining trends and drawing conclusions from data. Moreover, they offer a structured approach to problem-solving, where parameters are clearly defined, and relationships between different variables are mathematically outlined.
By systematically applying formulas and methods, mathematical modeling aids in transforming raw data into actionable insights.

Cross-Multiplication

Cross-multiplication is a fundamental technique used to solve proportions. It stems from the equality of two ratios, allowing us to find unknown quantities. This technique was applied to calculate the total number of voters in Seanoa City.

• We started with the given proportion: \(\frac{190}{400} = \frac{3,120}{x}\).
• Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other.

In our example, we performed: \(190x = 3,120 \times 400\). By solving for \(x\), we determined the total likely voter size.
Cross-multiplication is particularly effective for solving problems where relationships between different quantities are proportional, ensuring correct and straightforward computations.

Proportional Reasoning

Proportional reasoning allows us to understand how quantities relate to each other. When one quantity changes, we can predict how another will change, provided the relationship is proportional. In the case of our problem, we knew the proportion of voters favoring Candidate A in the sample.

• This connection helped us scale the sample data to estimate the number of supporters across the entire city.
• The proportion \(\frac{190}{400}\) served as a small-scale model of the wider population, revealing the nature of proportional relationships.

Understanding how to apply proportional reasoning can be powerful in many fields, such as physics, economics, and daily life. It simplifies complex problems by reducing them to a series of equivalent, smaller, and more manageable problems.
So, by using proportions, we can efficiently estimate and make sense of real-life scenarios, enhancing decision-making and problem-solving abilities.