Problem 42 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Mortality actuarial tables in the United States were revised in 2001, the fourth time since 1858 . Based on the new life insurance mortality rates, \(1 \%\) of 60 -yr-old men, \(2.6 \%\) of 70 -yr-old men, \(7 \%\) of 80 -yr-old men, \(18.8 \%\) of 90 -yr-old men, and \(36.3 \%\) of 100 -yr-old men would die within a year. The corresponding rates for women are \(0.8 \%, 1.8 \%, 4.4 \%, 12.2 \%\), and \(27.6 \%\), respectively. Express this information using a \(2 \times 5\) matrix.

Step-by-Step Solution

Answer

The 2x5 mortality matrix representing the given mortality rates for men and women of different age groups is: \[ \begin{pmatrix} 1 & 2.6 & 7 & 18.8 & 36.3 \\ 0.8 & 1.8 & 4.4 & 12.2 & 27.6 \end{pmatrix} \]

Step 1: Identify the Rows and Columns

Our matrix will have 2 rows, one for men and one for women, and 5 columns representing the age groups: 60 years old, 70 years old, 80 years old, 90 years old, and 100 years old.

Step 2: Fill in the Matrix Entries

Now, we will fill in the matrix entries with the given mortality rates. For men, we have: - 60 years old: \(1\%\) - 70 years old: \(2.6\%\) - 80 years old: \(7\%\) - 90 years old: \(18.8\%\) - 100 years old: \(36.3\%\) For women, we have: - 60 years old: \(0.8\%\) - 70 years old: \(1.8\%\) - 80 years old: \(4.4\%\) - 90 years old: \(12.2\%\) - 100 years old: \(27.6\%\)

Step 3: Create the 2x5 Matrix

Now that we have the mortality rates for all the age groups, we can create our 2x5 matrix. Mortality Matrix: \[ \begin{pmatrix} 1 & 2.6 & 7 & 18.8 & 36.3 \\ 0.8 & 1.8 & 4.4 & 12.2 & 27.6 \end{pmatrix} \] The resulting 2x5 mortality matrix has the mortality rates for 60-yr-old to 100-yr-old men in the first row and the rates for women in the second row.

Key Concepts

Mortality rates

Mortality rates are statistical measures that indicate the frequency at which death occurs in a particular population during a specific time period. These rates are often expressed as percentages, representing the proportion of individuals in an age group who are expected to die within a year.
Understanding mortality rates is crucial because it provides insights into the health and longevity of different groups over time. These rates are key indicators in healthcare planning, insurance policies, and public health initiatives.
In our exercise, the mortality rates for different age groups of men and women give a snapshot of the expected survival within a year. These values allow analysts to track trends and make crucial predictions about lifespan changes in populations.

Actuarial tables

Actuarial tables, also known as life tables, are used primarily in the insurance industry to evaluate the risk associated with insuring a person or group. These tables contain a variety of statistical data, including mortality rates, which help insurers determine the probability of a claim, life expectancy, and the premiums they should charge.
Each entry in an actuarial table provides detailed data on a specific age group, often including mortality rates, survival over different time intervals, and the number of people initially alive at that age. These tables serve as a standard for assessing life insurance policies and pension funds.
Our original exercise includes mortality rates from actuarial tables that help identify how many people from a specific age group are at risk of dying within a year. By utilizing these tables, actuaries can provide more accurate and fair insurance policies.

Matrix representation

Matrix representation is a powerful mathematical tool that allows for organizing data in a structured way. In applied mathematics, matrices are used to model and solve various complex problems efficiently. By transforming data into a matrix, calculations like additions, multiplications, and transformations become feasible and straightforward.
The mortality rates for men and women at various ages, when laid out in a matrix form, simplify data comparison and further analysis. Matrices support operations that could reveal trends and comparisons across datasets.

• The first row of our matrix represents mortality rates of men across different age groups: 60 to 100 years.
• Similarly, the second row covers mortality rates for women in the same age brackets.

This matrix format allows demographers and statisticians to compare the likelihood of survival or death between genders and within the same age brackets efficiently.

Demographic analysis

Demographic analysis involves studying a population's structure and changes over time, focusing on factors such as age, gender, birth rates, and mortality rates. It helps in understanding societal trends and the impacts of policy decisions.
In our context, demographic analysis uses mortality rates to predict future populations. Understanding these figures helps planners allocate resources more effectively, address healthcare needs, and forecast economic impacts due to aging populations.
Analyzing matrices of mortality data can reveal significant trends that might require policy intervention or adaptation. For instance, a noticeable increase in mortality rates for a specific group might suggest underlying health issues that need addressing. Such analyses guide strategic planning for national healthcare systems and social services.