Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Two hundred workers were asked: Would a better economy lead you to switch jobs? The results of the survey follow: $$ \begin{array}{lccccc} \hline & \text { Very } & \text { Somewhat } & \text { Somewhat } & \text { Very } & \text { Don't } \\ \text { Answer } & \text { likely } & \text { likely } & \text { unlikely } & \text { unlikely } & \text { know } \\ \hline \text { Respondents } & 40 & 28 & 26 & 104 & 2 \\ \hline \end{array} $$ If a worker is chosen at random, what is the probability that he or she a. Is very unlikely to switch jobs? b. Is somewhat likely or very likely to switch jobs?

Step-by-Step Solution

Answer

a. The probability that a worker chosen at random is very unlikely to switch jobs is \(0.52\) or \(52\%\). b. The probability that a worker chosen at random is either somewhat likely or very likely to switch jobs is \(0.34\) or \(34\%\).

Step 1: a. Probability of very unlikely to switch jobs

From the survey data, we know that 104 workers out of 200 are very unlikely to switch jobs. To find the probability that a worker chosen at random is very unlikely to switch jobs, we will divide the number of workers in this category by the total number of workers surveyed. \(P(\text{Very Unlikely}) = \frac{\text{Number of Very Unlikely Workers}}{\text{Total Number of Workers}}\)

Step 2: a. Calculation

Plug in the numbers from the survey: \(P(\text{Very Unlikely}) = \frac{104}{200} = 0.52\) So, there is a probability of 0.52 or 52% that a worker chosen at random is very unlikely to switch jobs.

Step 3: b. Probability of somewhat likely or very likely to switch jobs

From the survey, we know that 40 workers are very likely and 28 workers are somewhat likely to switch jobs. We will find the probability of a worker chosen at random to be either somewhat likely or very likely to switch jobs by adding the probability of these two categories together. \(P(\text{Somewhat Likely or Very Likely}) = P(\text{Somewhat Likely}) + P(\text{Very Likely})\) To find the probabilities for each category, we will use the same approach as before: \(P(\text{Somewhat Likely}) = \frac{\text{Number of Somewhat Likely Workers}}{\text{Total Number of Workers}}\) \(P(\text{Very Likely}) = \frac{\text{Number of Very Likely Workers}}{\text{Total Number of Workers}}\)

Step 4: b. Calculation

Plug in the numbers from the survey: \(P(\text{Somewhat Likely}) = \frac{28}{200} = 0.14\) \(P(\text{Very Likely}) = \frac{40}{200} = 0.20\) Now, add these probabilities together: \(P(\text{Somewhat Likely or Very Likely}) = 0.14 + 0.20 = 0.34\) So, there is a probability of 0.34 or 34% that a worker chosen at random is either somewhat likely or very likely to switch jobs.

Key Concepts

Survey Data Analysis

Understanding survey data analysis is crucial when dealing with information gathered through surveys. When conducting a survey, questions are presented to a defined group of individuals to collect specific data. In this case, 200 workers were surveyed to ascertain their likelihood of switching jobs if the economy improved.
The data can be classified into several categories based on responses, such as "Very likely," "Somewhat likely," "Somewhat unlikely," "Very unlikely," and "Don't know."
Analyzing this data involves interpreting the responses, counting the number of respondents in each category, and using this information to calculate probabilities. It is important to handle survey data correctly to ensure that the insights drawn are accurate and meaningful. This involves clean data collection, correct categorization, and accurate counting and aggregation of respondents in each category.

Probability Calculation

Probability calculation is a key component of analyzing survey data. Probability quantifies the likelihood of an event occurring. To calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes.
For example, if you need to find the probability of a worker being "very unlikely" to switch jobs, you count the workers in the "Very unlikely" category and divide this by the total number of workers surveyed. In this exercise, the probability of a worker being very unlikely to switch jobs is calculated as follows:

• The number of workers who are very unlikely is 104.
• The total number of workers surveyed is 200.

Thus, the probability is calculated with the formula: \[ P(\text{Very Unlikely}) = \frac{104}{200} = 0.52 \text{ or 52\%} \]
Probability calculations help in understanding the likelihood and drawing insights into how the group in question may behave under specific circumstances.

Random Selection Probability

The concept of random selection probability is prevalent in statistics and involves selecting an individual from a population in such a way that every individual has an equal chance of being selected. In the case of our survey example, drawing a worker randomly means each worker has an equal 1 in 200 chance of being chosen.
When calculating probabilities under random selection, it's crucial to ensure that selection is truly random and unbiased. Each individual should have an identical probability of selection regardless of prior observations.
In our exercise, finding the probability of a randomly selected worker being either "somewhat likely" or "very likely" to switch jobs involves calculating separate probabilities for each event and then summing them:

• Calculate \(P(\text{Somewhat Likely}) = \frac{28}{200} = 0.14\)
• Calculate \(P(\text{Very Likely}) = \frac{40}{200} = 0.20\)
• Add those probabilities to find the combined probability: \[P(\text{Somewhat Likely or Very Likely}) = 0.14 + 0.20 = 0.34\]

This means there's a 34% chance that a randomly chosen worker is either somewhat likely or very likely to switch jobs if the economy improves.