Question

A survey to determine the availability of flextime schedules in the California workplace provided the following information for 220 firms located in two California cities. $$\begin{array}{cccc}\multicolumn{4}{c} {\text { Flextime Schedule }} \\\\\hline \text { City } & \text { Available } & \text { Not Available } & \text { Total } \\\\\hline A & 39 & 75 & 114 \\\B & 25 & 81 & 106 \\\\\hline \text { Totals } & 64 & 156 & 220\end{array}$$ A company is selected at random from this pool of 220 companies. a. What is the probability that the company is located in city \(A\) ? b. What is the probability that the company is located in city \(B\) and offers flextime work schedules? c. What is the probability that the company does not have flextime schedules? d. What is the probability that the company is located in city \(B\), given that the company has flextime schedules available?

Short answer

The probability that a randomly selected company is in City A is 57/110. b. What is the probability that a randomly selected company is in City B and has a flextime work schedule? The probability that a randomly selected company is in City B and has a flextime work schedule is 5/44. c. What is the probability that a randomly selected company does not have a flextime work schedule? The probability that a randomly selected company does not have a flextime work schedule is 39/55. d. What is the probability that a randomly selected company is in City B, given that it has a flextime work schedule? The probability that a randomly selected company is in City B, given that it has a flextime work schedule is 25/176.

Step-by-step solution

a. Probability of company in City A

To determine the probability of a company being in city A, we can use the formula: $$P(\text{City A}) = \frac{\text{Number of Companies in City A}}{\text{Total Number of Companies}}$$ From the provided table, we can see that there are 114 companies in City A out of a total of 220 companies. Therefore the probability is: $$P(\text{City A}) = \frac{114}{220} = \frac{57}{110}$$

b. Probability of company in City B with flextime schedules

To determine the probability of a company being in City B and having a flextime work schedule, we can use the formula: $$P(\text{City B and Flextime}) = \frac{\text{Number of Companies in City B with Flextime Schedules}}{\text{Total Number of Companies}}$$ From the provided table, we can see that there are 25 companies in City B with flextime schedules out of a total of 220 companies. Therefore the probability is: $$P(\text{City B and Flextime}) = \frac{25}{220} = \frac{5}{44}$$

c. Probability of company without flextime schedules

To determine the probability of a company without flextime schedules, we can use the formula: $$P(\text{No Flextime}) = \frac{\text{Number of Companies without Flextime Schedules}}{\text{Total Number of Companies}}$$ From the provided table, we can see that there are 156 companies without flextime schedules out of a total of 220 companies. Therefore the probability is: $$P(\text{No Flextime}) = \frac{156}{220} = \frac{39}{55}$$

d. Probability of company in City B given flextime schedules

To determine the probability of a company being in City B given that the company has flextime schedules, we can use the conditional probability formula: $$P(\text{City B}|\text{Flextime}) = \frac{P(\text{City B and Flextime})}{P(\text{Flextime})}$$ We already found the probability of a company being in City B with flextime schedules (part b) as \(\frac{5}{44}\). Now we need to find the probability of a company having a flextime schedule, which can be calculated using the formula: $$P(\text{Flextime}) = \frac{\text{Number of Companies with Flextime Schedules}}{\text{Total Number of Companies}}$$ From the provided table, we can see that there are 64 companies with flextime schedules out of a total of 220 companies. Therefore the probability is: $$P(\text{Flextime}) = \frac{64}{220} = \frac{16}{55}$$ Now we can find the conditional probability: $$P(\text{City B}|\text{Flextime}) = \frac{\frac{5}{44}}{\frac{16}{55}} = \frac{5 \cdot 55}{44\cdot 16} = \frac{25}{176}$$

Key concepts

Conditional Probability

Conditional probability is a key concept in probability theory that allows us to compute the probability of an event occurring given that another event has already occurred. We use it to find how likely a certain condition affects an outcome. This can be particularly useful when we have additional information that influences the likelihood of events.

In the context of the exercise, when we want to find the probability of a company being located in City B, given that it offers flextime schedules, we apply conditional probability. We are interested in the likelihood that a company is from City B, knowing that it already offers flextime schedules. We use the notation \( P(\text{City B}|\text{Flextime}) \), where the vertical bar \(|\) signifies "given that." The formula we use is:

\[ P(\text{City B}|\text{Flextime}) = \frac{P(\text{City B and Flextime})}{P(\text{Flextime})} \]

This formula requires two pieces of information:

• The joint probability of being both in City B and offering flextime schedules, which we calculate by dividing the number of City B companies with flextime by the total number of companies.
• The probability of any company offering flextime, calculated by the number of all flextime companies divided by the total group size.

Conditional probability thus allows us to zoom in on a specific scenario, sharpening our understanding of the dynamics between related events.

Flextime Schedules

Flextime schedules refer to working arrangements that allow employees some flexibility in their work hours within certain limits. Instead of the traditional 9-to-5 model, flextime offers employees the opportunity to choose start and end times that better fit their personal and professional needs.

In the provided survey data, we are evaluating how prevalent flextime schedules are among firms in two Californian cities, City A and City B. Such data can inform us about the adoption of flextime in different regions, indicating possible regional preferences or economic conditions that favor flexible work arrangements.

Organizations may offer flextime to improve:

• Work-life balance, allowing employees to attend to personal matters while maintaining productivity.
• Job satisfaction, increasing morale and retention by offering more autonomy.
• Company productivity, as employees work at their most productive times.

The data analyzed helps highlight the specifics of where and how versatile these schedules are used and their possible impacts.

Survey Data Analysis

Survey data analysis refers to the methods used to collect, interpret, and present data obtained from surveys. It is an essential process for transforming raw data into meaningful information that can be used for making informed decisions.

In the exercise, survey data from 220 firms is analyzed to understand the distribution and frequency of flextime schedules in different cities. This type of analysis involves computing probabilities to quantify trends and patterns, making it easier to draw conclusions and identify insights.

The steps for analyzing survey data generally include:

• Collecting data through structured surveys that target specific information.
• Organizing and summarizing the collected data, often in tabular form like the table provided.
• Calculating probabilities and other statistical measures to identify correlations, like the degree to which city location might influence the availability of flextime schedules.
• Interpreting results to provide actionable insights, such as understanding geographical differences or employer trends.

Analysis of survey data allows companies and other stakeholders to assess current practices, predict future trends, and strategically plan for adjustments in policy and operations.