Problem 30 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

STUDENT DROPOUT RATE Data released by the Department of Education regarding the rate (percentage) of ninth-grade students who don't graduate showed that, out of 50 states, 12 states had an increase in the dropout rate during the past 2 yr. 15 states had a dropout rate of at least \(30 \%\) during the past 2 yr. 21 states had an increase in the dropout rate and/or a dropout rate of at least \(30 \%\) during the past 2 yr. a. How many states had both a dropout rate of at least \(30 \%\) and an increase in the dropout rate over the 2 -yr period? b. How many states had a dropout rate that was less than \(30 \%\) but that had increased over the 2 -yr period?

Step-by-Step Solution

Answer

a. 6 states had both a dropout rate of at least 30% and an increase in the dropout rate over the 2-year period. b. 6 states had a dropout rate less than 30% but experienced an increase over the 2-year period.

Step 1: Define sets and variables

Let `A` be the set of states that had an increase in the dropout rate during the past 2 years. Let `B` be the set of states that had a dropout rate of at least 30% during the past 2 years. We are given: - |A| = 12 - |B| = 15 - |A ∪ B| = 21 We are asked to find: a. |A ∩ B| b. |A| - |A ∩ B|

Step 2: Find the intersection of sets A and B

We will use the following set theory formula: |A ∪ B| = |A| + |B| - |A ∩ B| Calculate |A ∩ B|: |A ∩ B| = |A| + |B| - |A ∪ B| = 12 + 15 - 21 = 6

Step 3: Calculate the number of states with a dropout rate less than 30% but an increase over the 2-year period

We now find |A| - |A ∩ B|: |A| - |A ∩ B| = 12 - 6 = 6

Step 4: Answer the questions

a. The number of states that had both an increase in the dropout rate over the 2-year period and a dropout rate of at least 30% is 6. b. The number of states that had a dropout rate less than 30% but had an increase over the 2-year period is 6.

Key Concepts

Union and Intersection of Sets

Set theory is foundational in mathematics, and two of its most basic operations are the union and intersection of sets. The union of sets, denoted by the symbol \( \cup \), combines all the distinct elements from the contributing sets. For instance, if we have two sets, \(A\) and \(B\), then \(A \cup B\) would include every element that is in \(A\), \(B\), or both.

The intersection of sets, represented by the symbol \( \cap \), is the set containing elements common to all the contributing sets. So \(A \cap B\) includes only the elements that are in both \(A\) and \(B\). Applying these concepts to real-world problems, such as analyzing student dropout rates, helps us extract meaningful insights from data.

Cardinality of Sets

The cardinality of a set, denoted \( |A| \), is a measure of the 'number of elements' in the set \(A\). Cardinality can be thought of as the strength or size of a set. In applied mathematics and data analysis, understanding the cardinality of sets is important for quantifying concepts. For instance, when examining student dropout rates, the cardinality of the set representing states with an increase in dropout rates gives us a direct count of how many states are experiencing this issue.

Applied Mathematics

Applied mathematics involves utilizing mathematical methods and theories to solve practical problems from various disciplines such as science, engineering, business, and more. It plays a crucial role in enabling researchers and analysts to model real-world scenarios, predict outcomes, and make informed decisions. Add to this, set theory in mathematics is a tool often used in applied mathematics to manage and interpret data, as seen in exercises analyzing student dropout rates where set operations help to clarify the relationships between different data groups.

Student Dropout Rate Analysis

Analyzing student dropout rates is critical for educational institutions and policymakers in understanding and addressing educational challenges. By applying set theory to this problem, analysts can identify patterns and intersections among different data sets, such as states with increased dropout rates (\(A\)) and those with dropout rates above a certain threshold (\(B\)). Breaking down complex data into sets, and using concepts like union and intersection, helps to visualize and solve issues by finding the exact number of states falling into each category. This leads to actionable insights, which can inform strategies to improve student retention.