Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The property damage claim frequencies per 100 cars in Massachusetts in the years 2000,2001 , and 2002 are \(6.88,7.05\), and \(7.18\), respectively. The corresponding claim frequencies in the United States are 4.13, \(4.09\), and \(4.06\), respectively. Express this information using \(2 \times 3\) matrix.

Step-by-Step Solution

Answer

The given data for property damage claim frequencies per 100 cars in Massachusetts and the United States for the years 2000, 2001, and 2002 can be represented by a \(2 \times 3\) matrix as: \[ \begin{bmatrix} 6.88 & 7.05 & 7.18 \\ 4.13 & 4.09 & 4.06 \\ \end{bmatrix} \]

Step 1: |Year|

|Massachusetts|

Step 2: |United States| |-|-|-|-|-|-| |

|2000|

Step 3: |6.88|

|4.13| |

Step 4: |2001|

|7.05|

Step 5: |4.09| |

|2002|

Step 6: |7.18|

|4.06| Now, let's create a 2x3 matrix.

Step 7: Matrix Creation- Step 1:

Write down the matrix with 2 rows and 3 columns: \[ \begin{bmatrix} - & - & - \\ - & - & - \\ \end{bmatrix} \]

Step 8: Matrix Creation- Step 2:

Fill the first row with the damage claim frequencies in Massachusetts for each year: \[ \begin{bmatrix} 6.88 & 7.05 & 7.18 \\ - & - & - \\ \end{bmatrix} \]

Step 9: Matrix Creation- Step 3:

Fill the second row with the damage claim frequencies in the United States for each year: \[ \begin{bmatrix} 6.88 & 7.05 & 7.18 \\ 4.13 & 4.09 & 4.06 \\ \end{bmatrix} \] The final 2x3 matrix representing the property damage claim frequencies per 100 cars in Massachusetts and the United States for the years 2000, 2001, and 2002 is: \[ \begin{bmatrix} 6.88 & 7.05 & 7.18 \\ 4.13 & 4.09 & 4.06 \\ \end{bmatrix} \]

Key Concepts

Applied Mathematics

Applied mathematics is a branch of mathematics that deals with mathematical methods that are used in the application of mathematical knowledge to other domains. It connects the abstract world of numbers and formulas to the tangible realities we face in everyday life, such as economics, engineering, and physical sciences.

For example, in the given exercise, we use applied mathematics to quantify and organize the property damage claim frequencies for automobiles in Massachusetts and the United States over a series of years. These numerical values could then be analyzed to make decisions in economic policies or insurance rates.

In the context of the exercise, a real-world phenomenon is measured, recorded, and translated into a mathematical format that can facilitate further analysis. This demonstrates one of the many ways applied mathematics helps us solve concrete problems using abstract mathematical concepts.

Matrix Representation

Matrix representation is a powerful mathematical tool used to organize data efficiently and to perform a variety of calculations in several fields such as physics, computer science, and statistics. A matrix is essentially a rectangular array of numbers arranged in rows and columns. The numbers within a matrix can represent data or mathematical operations.

In the related exercise, we see a 2x3 matrix being used to represent claim frequencies for two different regions over three separate years. Each element within this matrix is a piece of data corresponding to a specific location and time.

Understanding Matrix Dimensions

In the example given, the dimension of the matrix is 2x3 because there are two rows – representing the two different geographical locations – and three columns – representing the different years. The matrix organizes this information in a clear, concise way that allows for easy comparison and analysis of the data.

Furthermore, matrix representation facilitates operations such as addition, subtraction, and multiplication of matrices, which are crucial in solving more complex applied mathematics problems, particularly in linear algebra and data analysis.

Data Analysis

Data analysis involves inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, drawing conclusions, and supporting decision-making. In our daily lives, we are surrounded by data—it shapes our understanding of the world and informs our decisions.

Using the matrix created in the exercise, one could perform various types of data analysis, such as trend analysis over the years for car damage claim frequencies. By analyzing this matrix, we can compare the frequencies between different regions and observe whether they are increasing or decreasing.

Matrices are invaluable in data analysis because they provide a structured way to manage multidimensional data. For instance, matrices can be manipulated algebraically and used in conjunction with other matrices to model complex relationships between variables. With the advancement of computing technologies, matrix operations are fundamental in performing large-scale data analysis efficiently. This is why matrices are at the heart of many data analysis techniques and statistical computing methods.