Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a Los Angeles Times poll of 1936 California residents conducted in February 2004 , the following question was asked: Do you favor or oppose an amendment to the U.S. Constitution barring same-sex marriage? The following results were obtained: $$ \begin{array}{lccc} \hline \text { Opinion } & \text { Favor } & \text { Oppose } & \text { Don't know } \\ \hline \text { Respondents } & 910 & 891 & 135 \\ \hline \end{array} $$ Determine the empirical probability distribution associated with these data.

Step-by-Step Solution

Answer

The empirical probability distribution associated with these data is as follows: - Favor: Approximately \(46.98\%\) - Oppose: Approximately \(46.01\%\) - Don't know: Approximately \(6.98\%\)

Step 1: Find the total number of respondents

First, let's find the total number of respondents in the poll. We can do this by summing the number of respondents for each opinion: Total respondents \(= 910 + 891 + 135 = 1936\)

Step 2: Determine the probability of each opinion

Now we will find the probability associated with each opinion. We can do this by dividing the number of respondents with that opinion by the total number of respondents. 1. Probability of Favor: \(P(Favor) = \frac{\text{Number of respondents favoring}}{\text{Total number of respondents}} = \frac{910}{1936}\) 2. Probability of Oppose: \(P(Oppose) = \frac{\text{Number of respondents opposing}}{\text{Total number of respondents}} = \frac{891}{1936}\) 3. Probability of Don't know: \(P(Don't \,know) = \frac{\text{Number of respondents with 'don't know' opinion}}{\text{Total number of respondents}} = \frac{135}{1936}\)

Step 3: Calculate the probabilities

Now we will calculate the probabilities for each opinion using the formulas obtained previously. 1. Probability of Favor: \(P(Favor) = \frac{910}{1936} \approx 0.4698\) 2. Probability of Oppose: \(P(Oppose) = \frac{891}{1936} \approx 0.4601\) 3. Probability of Don't know: \(P(Don't \,know) = \frac{135}{1936} \approx 0.0698\)

Step 4: Empirical probability distribution

Based on our calculations, we can write the empirical probability distribution as follows: $$\begin{cases} P(Favor) \approx 0.4698 \\ P(Oppose) \approx 0.4601 \\ P(Don't \,know) \approx 0.0698 \end{cases}$$ Therefore, the empirical probability distribution associated with these data is: - Favor: Approximately \(46.98\%\) - Oppose: Approximately \(46.01\%\) - Don't know: Approximately \(6.98\%\)

Key Concepts

Probability Calculation

Probability is a fundamental concept in statistics that allows us to quantify the likelihood of different outcomes. In this exercise, we are tasked with calculating empirical probabilities from survey data. Empirical probability is a type of probability derived from actual data, in contrast to theoretical probability, which is based on expected outcomes.
To calculate the probability of an event from our data, we use the formula:

• \( P( ext{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

In this context, each "event" corresponds to an opinion about the proposed constitutional amendment in the survey. For example, to calculate the probability that a surveyed resident favors the amendment, we divide the number of respondents who favor it (910) by the total number of respondents (1936).
The result is \( P(Favor) = \frac{910}{1936} \approx 0.4698 \). This means there's approximately a 46.98% chance that a randomly selected responder from the survey favors the amendment. By understanding these calculations, we gain deeper insights into public opinion as reflected by the data.

Data Analysis

Data analysis is all about making sense of numbers collected during surveys or experiments, which in turn prompts informed decisions and understanding. The survey results given here are numerical data that reflect the opinions of respondents about a constitutional amendment regarding same-sex marriage.
We started our analysis by summing up the responses in each category—Favor, Oppose, and Don't know—to find the total sample size. This step is crucial because it lays the groundwork for everything that follows in data analysis, allowing us to specify what fraction of the sample corresponds to each opinion.
In this analysis, we determined the frequency of each category:

• Favor: 910 responses
• Oppose: 891 responses
• Don't know: 135 responses

Knowing these frequencies helps us calculate empirical probabilities by representing them as percentages of the total. This is a vital aspect of analyzing survey data, as it portrays the overall sentiment or trend within the collected samples, giving a clear snapshot of public opinion on the issue at the time of the survey.

Survey Results

Survey results are invaluable for understanding trends and opinions in a population. Here, the survey conducted with 1936 California residents provides a rich dataset for exploring public attitudes towards a specific constitutional amendment.
Surveys collect responses from a subset of a population, allowing for estimates of opinions or behaviors without needing to ask everyone. The results give us a snapshot of how many people support, oppose, or are undecided about the amendment:

• Favor: Approximately 46.98%
• Oppose: Approximately 46.01%
• Don't know: Approximately 6.98%

These percentages are calculated from the responses and portray the empirical probability distribution, which aids in understanding the overall landscape of opinion.
The high percentages of "Favor" and "Oppose" indicate a closely divided public opinion on the matter, while a smaller "Don't know" percentage suggests fewer undecided respondents. Such data is crucial for policymakers, advocacy groups, and researchers when considering areas of public policy or societal interest, as they provide a direct view into what the majority of people think and feel on important issues.