Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The 120 consumers of Exercise 19 were also asked about their buying preferences concerning another product that is sold in the market under three labels. The results were 12 buy only those sold under label A. 25 buy only those sold under label B. 26 buy only those sold under label C. 15 buy only those sold under labels \(\mathrm{A}\) and \(\mathrm{B}\). 10 buy only those sold under labels \(\mathrm{A}\) and \(\mathrm{C}\). 12 buy only those sold under labels \(\mathrm{B}\) and \(\mathrm{C}\). 8 buy the product sold under all three labels. How many of the consumers surveyed buy the product sold under a. At least one of the three labels? b. Labels A and B but not C? c. Label \(\mathrm{A}\) ? d. None of these labels?

Step-by-Step Solution

Answer

a. At least one of the three labels: 63 consumers. b. Labels A and B but not C: 7 consumers. c. Label A: 29 consumers. d. None of these labels: 57 consumers.

Step 1: Understand the given data

First, let's represent the given data with the following notations: - A: Consumers who buy products under Label A only. - B: Consumers who buy products under Label B only. - C: Consumers who buy products under Label C only. - AB: Consumers who buy products under Labels A and B only. - AC: Consumers who buy products under Labels A and C only. - BC: Consumers who buy products under Labels B and C only. - ABC: Consumers who buy products under all the three labels. Based on the provided information, we have: A = 12, B = 25, C = 26, AB = 15, AC = 10, BC = 12, ABC = 8

Step 2: Find the total number of consumers buying at least one label

Using PIE, we calculate the total number of consumers buying at least one of the three labels (A, B, or C). Total Consumers buying at least one label = A + B + C - (AB + AC + BC) + ABC Total = 12 + 25 + 26 - (15 + 10 + 12) + 8 Total = 63 a. At least one of the three labels: 63 consumers.

Step 3: Find the number of consumers buying Labels A and B but not C

We need to find consumers who buy products under labels A and B but not under label C. These consumers are the ones counted in the AB group but not in AC or BC group. Consumers buying A and B but not C = AB - ABC Result = 15 - 8 Result = 7 b. Labels A and B but not C: 7 consumers.

Step 4: Find the total number of consumers buying Label A

We need to find the total number of consumers who buy products under label A. These consumers include those who buy A only, A and B, and A and C. Consumers buying Label A = A + AB + AC - ABC Result = 12 + 15 + 10 - 8 Result = 29 c. Label A: 29 consumers.

Step 5: Find the number of consumers not buying any of these labels

We need to find the number of consumers who don't buy any of these labels (A, B, or C). To find this, we can subtract the total number of consumers buying at least one label from the total number of consumers surveyed. Consumers not buying any label = Total Consumers - Consumers buying at least one label Result = 120 - 63 Result = 57 d. None of these labels: 57 consumers.

Key Concepts

Principle of Inclusion-Exclusion

The Principle of Inclusion-Exclusion (PIE) is a fundamental method in set theory used to calculate the size of the union of multiple sets. It's particularly useful when dealing with overlapping sets, ensuring we accurately include or exclude elements that belong to multiple sets simultaneously. Here's the basic idea with three sets, A, B, and C:

• Add the sizes of the three sets: \( |A| + |B| + |C| \).
• Subtract the sizes of the pairwise intersections because these elements have been counted twice: \( -(|A \cap B| + |A \cap C| + |B \cap C|) \).
• Add back the size of the intersection of all three sets since these elements have been subtracted one too many times: \( + |A \cap B \cap C| \).

This formula ensures every element is considered the correct number of times. In the original problem, we used PIE to determine how many consumers buy at least one label. This kind of calculation is typical in surveys or market analyses where individuals can belong to more than one category.

Survey Analysis

Survey analysis involves collecting and examining data from a particular group of participants to discover insights or trends. In our case, it focuses on consumers and their buying preferences across different product labels. Surveys often involve asking participants about their choices and categorizing the responses. This approach allows businesses to understand trends, preferences, and even forecast demand. In this exercise, consumers were asked about their preferences concerning products sold under three different labels. From the survey, data is categorized into distinct groups:

• Consumers who buy only from one specific label.
• Consumers who buy from combinations of labels.
• Consumers who buy from all or no labels.

Analyzing this data helps businesses determine how many people prefer certain combinations of labels and derive insights into overall purchasing behaviors.

Consumer Behavior Analysis

Consumer behavior analysis involves studying how individuals make decisions to spend their available resources, such as time and money, on consumption-related items. For businesses, understanding these behaviors can provide valuable insights. In this particular exercise, consumer behavior is observed based on the labels they prefer for a particular product. By analyzing such data, businesses can determine how varied labels influence buying decisions. Key aspects considered in consumer behavior analysis include:

• Why consumers prefer certain labels over others.
• The influence of brand combinations on purchasing decisions.
• Trends in consumer choices, which can inform marketing strategies.

By understanding which labels are more popular, businesses can tailor their marketing efforts and product distributions to fit consumer preferences better. The analysis can also highlight potential areas for product improvement or rebranding to align with consumer desires.