Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a survey of 2000 adults 18 yr and older conducted in 2007 , the following question was asked: Is your family income keeping pace with the cost of living? The results of the survey follow: $$ \begin{array}{lcccc} \hline & \begin{array}{c} \text { Falling } \\ \text { behind } \end{array} & \begin{array}{c} \text { Staying } \\ \text { even } \end{array} & \begin{array}{c} \text { Increasing } \\ \text { faster } \end{array} & \begin{array}{c} \text { Don't } \\ \text { know } \end{array} \\ \hline \text { Respondents } & 800 & 880 & 240 & 80 \\ \hline \end{array} $$ Determine the empirical probability distribution associated with these data.

Step-by-Step Solution

Answer

The empirical probability distribution for the given data can be represented as follows: - Falling Behind: \(P(Falling Behind) = \frac{2}{5}\) - Staying Even: \(P(Staying Even) = \frac{11}{25}\) - Increasing Faster: \(P(Increasing Faster) = \frac{3}{25}\) - Don't Know: \(P(Don't Know) = \frac{1}{25}\)

Step 1: Understand the data given

From the data provided in the table, we can see that: - 800 respondents said their family income was Falling Behind the cost of living. - 880 respondents said their family income was Staying Even with the cost of living. - 240 respondents said their family income was Increasing Faster than the cost of living. - 80 respondents said they Don't Know. Total number of respondents = 2000.

Step 2: Calculate probabilities for each category

We will now calculate the empirical probability for each category by dividing the number of respondents in each category by the total number of respondents (2000). - Probability of Falling Behind = 800 / 2000 - Probability of Staying Even = 880 / 2000 - Probability of Increasing Faster = 240 / 2000 - Probability of Don't Know = 80 / 2000

Step 3: Simplify the probabilities

Let's simplify the probabilities we calculated in step 2 by dividing both the numerator and denominator by their greatest common divisor (GCD). - Probability of Falling Behind = 800 / 2000 = 2 / 5 - Probability of Staying Even = 880 / 2000 = 11 / 25 - Probability of Increasing Faster = 240 / 2000 = 3 / 25 - Probability of Don't Know = 80 / 2000 = 1 / 25

Step 4: Present the empirical probability distribution

The empirical probability distribution for the given data is as follows: - Falling Behind: P(Falling Behind) = 2 / 5 - Staying Even: P(Staying Even) = 11 / 25 - Increasing Faster: P(Increasing Faster) = 3 / 25 - Don't Know: P(Don't Know) = 1 / 25

Key Concepts

Survey Data Analysis

Survey data analysis is a powerful tool used to extract meaningful information from collected data. In the context of the exercise, survey data analysis helped in understanding how respondents feel about their family income in relation to the cost of living. Here, the survey data shows us responses across four categories: "Falling Behind," "Staying Even," "Increasing Faster," and "Don't Know." Each choice reflects different perspectives of respondents on their economic situation, giving us insights into broader economic sentiments during the survey time.
Before diving into probabilities, it is key to comprehend the raw data. This involves listing the number of responses per category:

• "Falling Behind": 800 respondents
• "Staying Even": 880 respondents
• "Increasing Faster": 240 respondents
• "Don't Know": 80 respondents

Awareness of the total number of respondents, which is 2000, is crucial. It helps in calculating the probabilities and understanding the relative size of each group within the whole population.

Probability Simplification

Once we have calculated initial probabilities, simplifying them helps in creating clearer and more interpretable results. Probability simplification involves expressing the probabilities in their simplest fractional form by reducing them.
For simplification, we calculate the Greatest Common Divisor (GCD) for the numerator and the denominator of each probability. Reducing the probability fractions, as seen in the exercise:

• Falling Behind: \( \frac{800}{2000} = \frac{2}{5} \)
• Staying Even: \( \frac{880}{2000} = \frac{11}{25} \)
• Increasing Faster: \( \frac{240}{2000} = \frac{3}{25} \)
• Don't Know: \( \frac{80}{2000} = \frac{1}{25} \)

Why simplify? Because simplified fractions are easier to compare and understand. They also make subsequent calculations more manageable. For example, knowing that the chance of "Falling Behind" is 2/5 gives immediate visual insights that it is a significant portion of the total probability distribution.

Empirical Probability Calculation

Empirical probability calculation involves determining the chance of an event based on observed data. It is the proportion of observed occurrences within a sample space. In this exercise, empirical probabilities are calculated for each response category by dividing the number of responses in that category by the total number of respondents (2000).
The steps for empirical probability calculation are:

• Identify the total number of outcomes, which is 2000 in this survey.
• Count occurrences of each survey response category.
• Calculate probabilities: Divide each category count by the total respondents.

The calculated empirical probabilities represent the likelihood of a randomly chosen respondent selecting each category. These values provide a data-based understanding of public sentiment regarding family income versus cost of living during 2007. By viewing these probabilities, one can infer and analyze broader trends or dissatisfaction levels within the surveyed population.