Problem 28 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a survey of 1000 eligible voters selected at random, it was found that 80 had a college degree. Additionally, it was found that \(80 \%\) of those who had a college degree voted in the last presidential election. whereas \(55 \%\) of the people who did not have a college degree voted in the last presidential election. Assuming that the poll is representative of all eligible voters, find the probability that an eligible voter selected at random a. Had a college degree and voted in the last presidential election. b. Did not have a college degree and did not vote in the last presidential election. c. Voted in the last presidential election. d. Did not vote in the last presidential election.

Step-by-Step Solution

Answer

The probabilities for each event are as follows: a. Having a college degree and voting in the last presidential election: \(P(A) = 0.064\) b. Not having a college degree and not voting in the last presidential election: \(P(B) = 0.414\) c. Voting in the last presidential election: \(P(C) = 0.57\) d. Not voting in the last presidential election: \(P(D) = 0.43\)

Step 1: Determine the number of college graduates and non-graduates

We are given that 80 out of the 1000 eligible voters have a college degree. So, in the sample: Number of college degree-holders: 80 Number of non-degree-holders: 1000 - 80 = 920

Step 2: Calculate the number of voters based on a degree

80% of college graduates voted in the last election. So, the number of degree-holders who voted is: (0.8) * 80 = 64 55% of non-degree holders voted in the last election. So, the number of non-degree holders who voted is: (0.55) * 920 = 506

Step 3: Calculate the probability of each event

a. Probability of having a college degree and voted in the last election: P(A) = \(\frac{\text{Number of degree-holders who voted}}{\text{Total number of eligible voters}}\) = \(\frac{64}{1000}\) b. Probability of not having a college degree and did not vote in the last election: Number of non-degree holders who did not vote: 920 - 506 = 414 P(B) = \(\frac{\text{Number of non-degree holders who did not vote}}{\text{Total number of eligible voters}}\) = \(\frac{414}{1000}\) c. Probability of voting in the last election: Total number of voters: 64 (degree-holders who voted) + 506 (non-degree holders who voted) = 570 P(C) = \(\frac{\text{Total number of voters}}{\text{Total number of eligible voters}}\) = \(\frac{570}{1000}\) d. Probability of not voting in the last election: Total number of non-voters: 1000 - 570 = 430 P(D) = \(\frac{\text{Total number of non-voters}}{\text{Total number of eligible voters}}\) = \(\frac{430}{1000}\)

Step 4: Present the final probabilities

We have calculated all the required probabilities for this exercise. Thus, we have: a. The probability of having a college degree and voting in the last presidential election: \(P(A) = \frac{64}{1000} = 0.064\) b. The probability of not having a college degree and not voting in the last presidential election: \(P(B) = \frac{414}{1000} = 0.414\) c. The probability of voting in the last presidential election: \(P(C) = \frac{570}{1000} = 0.57\) d. The probability of not voting in the last presidential election: \(P(D) = \frac{430}{1000} = 0.43\)

Key Concepts

College Degree

A college degree often represents advanced education and skill development. In statistics, measuring how many individuals have a college degree can indicate educational levels within a population. In our exercise, 80 out of 1000 eligible voters have a college degree.
This forms the basis for examining how education influences behaviors, like voting.
Education level is often linked with awareness and participation in civic duties, which can be critical when analyzing societal dynamics.
A difference in voter turnout between college graduates and non-graduates, shown in our example, suggests that higher education may correlate with higher civic engagement.

Voter Turnout

Voter turnout is a key indicator of political engagement within a population. It shows the percentage of eligible voters who participated in an election. Higher voter turnout is often seen as beneficial for democracy, indicating a healthy public interest in governance.
In the exercise, 80% of voters with a college degree participated in the last election, compared to 55% of those without a degree.
This difference highlights how education might influence the likelihood of voting.
Understanding such patterns can help in designing strategies to increase overall voter participation. Knowing who votes can inform how political messages or policies are directed.

Educational Attainment

Educational attainment refers to the highest level of education that individuals have completed. It's a vital measure in statistical surveys because it offers insights into the population's educational profile.
In our scenario, determining the number of voters with or without a degree provides data essential for understanding the broader educational achievement landscape.
This, in turn, explains not only how many people are educated at a certain level, but also how this education level might be affecting other areas like employment rates or civic activities, including voting.
By analyzing the relationship between education level and voter turnout, researchers can identify trends and correlations that matter for policy-making and educational reform.

Statistical Surveys

Statistical surveys are tools used to collect data from a predetermined group, giving a snapshot of the opinions, behavior or characteristics of that group.
Our exercise involved a survey of 1000 eligible voters. Such surveys are essential in understanding larger populations by sampling a smaller group that represents the whole.
The results, like those showing the rate of college degree holders who vote, can then be used to infer behaviors and tendencies of entire populations.
Good survey design is critical, involving clear objectives, random sampling, and consistent data gathering to ensure the findings are valid and reliable. These insights can shape societal understanding and inform decisions at various levels.