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Chapter 5: Problem 44

The market share of motorcycles in the United States in 2001 follows: Honda \(27.9 \%\), Harley-Davidson \(21.9 \%\), Yamaha \(19.2 \%\), Suzuki \(11.0 \%\), Kawasaki \(9.1 \%\), and others \(10.9 \%\). The corresponding figures for 2002 are \(27.6 \%, 23.3 \%, 18.2 \%\), \(10.5 \%, 8.8 \%\), and \(11.6 \%\), respectively. Express this information using a \(2 \times 6\) matrix. What is the sum of all the elements in the first row? In the second row? Is this expected? Which company gained the most market share between 2001 and \(2002 ?\)

Short Answer

Expert verified
The market share matrix is: \[ \begin{pmatrix} 27.9 & 21.9 & 19.2 & 11.0 & 9.1 & 10.9 \\ 27.6 & 23.3 & 18.2 & 10.5& 8.8 & 11.6 \\ \end{pmatrix} \] The sum of elements for each row is 100%, which is expected since the total market share must equal 100%. Between 2001 and 2002, Harley-Davidson gained the most market share with a 1.4% increase.

Step by step solution

01

Create the market share matrix

To create the matrix representing the market shares for the given companies in 2001 and 2002, we need to arrange the percentages in rows and columns. We can use the first row for 2001 and the second row for 2002. The columns can represent, in order, Honda, Harley-Davidson, Yamaha, Suzuki, Kawasaki, and others. The 2x6 matrix will be: \[ \begin{pmatrix} 27.9 & 21.9 & 19.2 & 11.0 & 9.1 & 10.9 \\ 27.6 & 23.3 & 18.2 & 10.5& 8.8 & 11.6 \\ \end{pmatrix} \]
02

Calculate the sum of the elements in each row

To find the sum of the elements in each row, simply add the percentages in each row: Sum of elements in the first row (2001): \(27.9 + 21.9 + 19.2 + 11.0 + 9.1 + 10.9 = 100\%\) Sum of elements in the second row (2002): \(27.6 + 23.3 + 18.2 + 10.5 + 8.8 + 11.6 = 100\%\)
03

Discuss whether the sum is expected

The sum being equal to 100% in each row is expected, as the total market share percentages must add up to 100% for each year.
04

Identifying the company with the most gained market share

To find out which company gained the most market share between 2001 and 2002, we need to calculate the difference in percentages for each company: - Honda: \(27.6\% - 27.9\% = -0.3\%\) - Harley-Davidson: \(23.3\% - 21.9\% = 1.4\%\) - Yamaha: \(18.2\% - 19.2\% = -1.0\%\) - Suzuki: \(10.5\% - 11.0\% = -0.5\%\) - Kawasaki: \(8.8\% - 9.1\% = -0.3\%\) - Others: \(11.6\% - 10.9\% = 0.7\%\) Based on these values, Harley-Davidson has gained the most market share between 2001 and 2002 with a 1.4% increase.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Market Share Analysis
Market share analysis helps us understand how various companies perform within an industry. Here, we are looking at motorcycle sales in the United States for the years 2001 and 2002. Each company, like Honda, Harley-Davidson, Yamaha, Suzuki, Kawasaki, and a category labeled as 'others', has a specific percentage showing its share of the market.

Analyzing market shares over time gives us insights into the business landscape. When listed in a matrix for each year, we can visually compare the changes or stability in market presence. This exercise involves summarizing data in a matrix format, which makes complex information clearer. Furthermore, identifying market leaders or those gaining ground over time provides valuable strategic insights.
Row and Column Sums
Working with matrices often involves calculating row and column sums. This is relevant in market share matrices, where each row represents a complete summary of a year's market share distribution.

For example, the sum of the row entries for market shares in 2001 and 2002 should equal to 100%. This reflects that all possible market shares for each year have been accounted for. Each company's share is a portion of this total. Checking row sums ensures that no part of the market is left unaccounted, which is crucial for accurate analysis.

Column sums would depict contributions by specific companies over the years, although this isn't performed in this example. Still, the idea of using sums can be important in analyzing total market contributions by individual players.
Percentages in Matrices
Using matrices to represent percentages is a practical way to handle data that deals with proportional information. Percentages offer intuitive insights, especially when the whole is fixed, as in 100% of the market.

In a matrix, each element can represent a percentage, such as the market share for a company in a given year. This is helpful because it keeps data organized, allowing for easy computation and comparison. Clearly structured data helps in identifying trends, such as increases or decreases in market shares over time.

When analyzing percentages in matrices, it's also essential to remain aware that changes in these figures can affect the perception of a company鈥檚 performance. A gain in percentage means a relative gain in market position, which is crucial for strategic planning and competitive analysis.

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Most popular questions from this chapter

(a) write each system of equations as a matrix equation and (b) solve the system of equations by using the inverse of the coefficient matrix. \(x_{1}+x_{2}+x_{3}+x_{4}=b_{1}\) \(x_{1}-x_{2}-x_{3}+x_{4}=b_{2}\) \(x_{2}+2 x_{3}+2 x_{4}=b_{3}\) \(x_{1}+2 x_{2}+x_{3}-2 x_{4}=b_{4}\) where (i) \(b_{1}=1, b_{2}=-1, b_{3}=4, b_{4}=0\) and (ii) \(b_{1}=2, b_{2}=8, b_{3}=4, b_{4}=-1\)

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Compute the indicated products. \(\left[\begin{array}{ll}0.1 & 0.9 \\ 0.2 & 0.8\end{array}\right]\left[\begin{array}{ll}1.2 & 0.4 \\ 0.5 & 2.1\end{array}\right]\)
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