Problem 53 - Precalculus: Mathematics for Calculus

Question

Education and lncome \(\quad\) A women's group takes a survey to determine the education and income of its members. Matrix \(A\) summarizes the proportions of members in various categories of years of postsecondary education and income. Matrix \(B\) shows the total number of members in each income category. (a) Calculate the product matrix \(A B\) (b) Interpret the entries of the matrix \(A B\) $$\begin{array}{rccc} & \text { Income level } \\ \hline \text { Less than } & \$ 50,000 & \$ 100,000 \\ \$ 50,000 & \text { to }\$ 100,000 & \text { or more } \end{array}$$ $$\begin{aligned}&\begin{array}{c}\text { NONE } \\ 1 \text { to } 4 \\\\\ \text { or more } 4 \end{array}\left[\begin{array}{r}0.75 & 0.10 & 0 \\ 0.25 & 0.70 & 0.70 \\ 0 & 0.20 & 0.30 \end{array}\right]=A\end{aligned}$$ $$\begin{aligned}&\qquad \qquad\qquad\quad\text { TOTAL }\\\&\begin{array}{c}\text { Less than } \$ 50,000 \\ \$ 50,000 \text { to } 100,000 \\\\\$ 100,000 \text { or more }\end{array}\left[\begin{array}{r}4 \\\20 \\\10\end{array}\right]=B\end{aligned}$$

Step-by-Step Solution

Answer

Matrix \(AB\) is \(\begin{bmatrix} 5 \\ 22 \\ 7 \end{bmatrix}\), indicating members' distribution across education levels.

Step 1: Understand the Problem

We are given matrices \(A\) and \(B\). Matrix \(A\) represents the proportion of members in various categories of education level and income, while matrix \(B\) provides the total members in each income category. Our task is to calculate the product of these two matrices.

Step 2: Setup the Matrices

Matrix \(A\) is a 3x3 matrix representing education levels with proportions for different income categories: \(\begin{bmatrix} 0.75 & 0.10 & 0 \ 0.25 & 0.70 & 0.70 \ 0 & 0.20 & 0.30 \end{bmatrix}\). Matrix \(B\) is a 3x1 matrix representing the total number of members in each income category: \(\begin{bmatrix} 4 \ 20 \ 10 \end{bmatrix}\).

Step 3: Calculate Matrix Product AB

To find \(AB\), multiply row elements of \(A\) by the corresponding elements of \(B\) and sum them for each row.1. First row: \(0.75\times4 + 0.10\times20 + 0\times10 = 3 + 2 = 5\).2. Second row: \(0.25\times4 + 0.70\times20 + 0.70\times10 = 1 + 14 + 7 = 22\).3. Third row: \(0\times4 + 0.20\times20 + 0.30\times10 = 0 + 4 + 3 = 7\).Thus, \(AB = \begin{bmatrix} 5 \ 22 \ 7 \end{bmatrix}\).

Step 4: Interpret the Result

The resulting product matrix \(AB = \begin{bmatrix} 5 \ 22 \ 7 \end{bmatrix}\) depicts the estimated number of members in each educational category based on income. This means there are approximately 5 members with 'NONE' education, 22 members with '1 to 4' years of postsecondary education, and 7 members with '4 or more' years, all within the given income segments.

Key Concepts

Matrix Interpretation

Matrix interpretation can be quite fascinating. In essence, when we have a matrix like Matrix \(A\), it holds proportions. Each entry in this matrix represents a segment of the data we're analyzing. This data may relate to real-world factors, such as education levels and income brackets in a survey.
For example, consider an entry in the matrix \(A\): 0.75 might indicate that 75% of surveyed members with no postsecondary education fall into a specific income category.
To interpret these matrices, always look at the context they describe. Ask yourself, "What does each number represent?"
Once you can answer that, you'll see how matrices transform data from abstract collections into tangible insights about populations and patterns.
Understand how each part of the data aligns with real-life categories, like education levels and income brackets. This alignment is what makes matrices a powerful tool for data interpretation.

Education and Income Survey

Surveys on education and income levels provide valuable information for understanding socio-economic dynamics. They can highlight how different levels of education might correlate with income levels.
In this context, matrix \(A\) shows us the survey results in terms of proportions of women surveyed, categorized by education level and income. This type of survey helps us identify potential trends, such as whether higher education correlates with higher income.
In conducting such a survey, we aim to gather specifics on:

• The distribution of education levels within different income brackets.
• The proportion of each educational category present in various income categories.

These insights allow organizations to make informed decisions that address educational needs and economic improvement opportunities in the community.

Proportions in Matrices

Proportions in matrices are all about how parts of the data relate to the whole. Entries in a proportion matrix, like \(A\), represent fractional values or percentages rather than absolute numbers. They tell us how an educational level is distributed across income categories according to the survey results.
The proportion matrix enables you to understand:

• How large a group of the surveyed people with no education earn less than \\(50,000.
• How many of those with '1 to 4 years' education earn between \\)50,000 to \$100,000.

By converting these proportions into actual population sizes through matrix multiplication, as shown in our exercise's solution, we derive actionable insights. Doing this allows you to draw conclusions and perform assessments that can support strategic decision-making, whether aiming for policy planning or educational programming.