Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a survey conducted in 2007 of 1402 workers 18 yr and older regarding their opinion on retirement benefits, the following data were obtained: 827 said that it was better to have excellent retirement benefits with a lower-than-expected salary, 477 said that it was better to have a higher-than-expected salary with poor retirement benefits, 42 said "neither," and 56 said "not sure." If a worker in the survey is selected at random, what is the probability that he or she answered that it was better to have a. Excellent retirement benefits with a lower-than-expected salary? b. A higher-than-expected salary with poor retirement benefits?

Step-by-Step Solution

Answer

The probability that a randomly selected worker prefers excellent retirement benefits with a lower-than-expected salary is approximately \(0.5906\) or \(59.06\%\). The probability that a randomly selected worker prefers a higher-than-expected salary with poor retirement benefits is approximately \(0.3402\) or \(34.02\%\).

Step 1: a. Probability of preferring excellent retirement benefits with a lower-than-expected salary

To find the probability of a random worker preferring excellent retirement benefits with a lower-than-expected salary, we need to divide the number of workers who chose this option by the total number of workers surveyed. Number of workers who prefer excellent retirement benefits with a lower-than-expected salary: 827 Total number of workers surveyed: 1402 probability = \(\frac{number\ of\ workers\ with\ the\ preference}{total\ number\ of\ workers\ surveyed}\) probability = \(\frac{827}{1402} \approx 0.5906\) The probability that a worker selected at random answered that it was better to have excellent retirement benefits with a lower-than-expected salary is approximately 0.5906 or 59.06%.

Step 2: b. Probability of preferring a higher-than-expected salary with poor retirement benefits

To find the probability of a random worker preferring a higher-than-expected salary with poor retirement benefits, we need to divide the number of workers who chose this option by the total number of workers surveyed. Number of workers who prefer a higher-than-expected salary with poor retirement benefits: 477 Total number of workers surveyed: 1402 probability = \(\frac{number\ of\ workers\ with\ the\ preference}{total\ number\ of\ workers\ surveyed}\) probability = \(\frac{477}{1402} \approx 0.3402\) The probability that a worker selected at random answered that it was better to have a higher-than-expected salary with poor retirement benefits is approximately 0.3402 or 34.02%.

Key Concepts

Statistical Analysis

Statistical analysis involves collecting data and using formulas to interpret that data effectively. In a survey, as in the example provided, it becomes a fundamental tool.
By using statistical methods, you can infer probabilities and draw meaningful conclusions.
When you calculate the probability, you determine the likelihood of a certain outcome happening in this specific data set.
The formula used is simple and can be applied to different survey scenarios:

• Total Number of Favorable Outcomes: This is the count of individuals who chose a specific preference in the survey.
  
• Total Number of Possible Outcomes: This represents the total number of respondents in the survey.
  

For our example, to find the respective probabilities for each choice, divide the number of workers with each preference by the total number of workers.
This is the basic principle of statistical analysis; it turns raw data into information you can use.
Analyzing such data helps organizations or researchers understand trends and behavior in specific populations.

Survey Data Interpretation

Survey data interpretation helps make sense of collected survey responses to gain insights, which can inform decisions or policy-making.
When looking at the probability results from our example survey, we start by examining the meaning behind the numbers.
In this case, understanding that 59.06% prefer excellent retirement benefits with a lower salary, while 34.02% prefer the opposite, might indicate the workers' priorities concerning financial security in retirement.
It tells us not only what respondents think but also guides future inquiries and actions.
Thinking about these results can help decide whether retirement plans need adjustments or if there’s a trend towards valuing immediate compensation over long-term benefits.

• Examine the different segments of the data
  
• Understand the implications of each segment and what it means in a broader context
  

Survey data interpretation allows for breaking down complex societal questions and finding out what motivates or concerns people.

Applied Mathematics

Applied mathematics is using mathematical methods by different fields such as science or engineering to solve real-world problems.
Probability calculation in surveys is one such application.
In applied mathematics, formulas transform data into actionable insights.
The ability to calculate probabilities allows researchers to predict potential outcomes and make better-informed decisions.
For example, knowing that a significant portion of the workforce values retirement benefits over immediate salary might not only help businesses but also impact broader economic policies.
Through mathematical equations, you convert numbers into probabilities. It involves the following steps:

• Identify the number of occurrences for a specific preference
  
• Determine the total sample size
  
• Use the formula for probability: \(\frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}\)
  

Applied mathematics not only helps individuals understand the likelihood of particular preferences but also bridges the gap between theoretical math and practical applications in society.