/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Data released by the Department ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 30

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?

Short Answer

Expert verified
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 had an increased dropout rate over the 2-year period.

Step by step solution

01

Organize the information

We are given the following information: 1. 12 states had an increase in the dropout rate. 2. 15 states had a dropout rate of at least 30%. 3. 21 states had either an increase in the dropout rate or a dropout rate of at least 30%. Let A represent the states with an increased dropout rate, and B represent the states with a dropout rate of at least 30%. We can represent this information in an organized table: | | A (Increase) | Not A (No increase) | Total | |-------------|--------------|--------------------|-------| | B (>=30%) | x | | 15 | | Not B (<30%)| | | | | Total | 12 | | 50 | #find the number of states that satisfy both conditions#
02

Apply the principle of inclusion and exclusion

Using the principle of inclusion and exclusion, we can find the number of states that had both a dropout rate of at least 30% and an increase in the dropout rate. This can be done by subtracting the total number of states that satisfy either one or both conditions from the individual sums, like so: |A 鈭� B| = |A| + |B| - |A 鈭� B| Where |A 鈭� B| is the number of states that satisfy both conditions (x). 21 = 12 + 15 - x Solving for x, we get: x = 12 + 15 - 21 x = 6 Thus, 6 states had both a dropout rate of at least 30% and an increase in the dropout rate over the 2-year period. This is the answer for part (a). Now we will fill our table with the new information: | | A (Increase) | Not A (No increase) | Total | |-------------|--------------|--------------------|-------| | B (>=30%) | 6 | | 15 | | Not B (<30%)| | | | | Total | 12 | | 50 | #find the number of states that only satisfy one of the conditions#
03

Complete the table and find the answer for part (b)

Next, we need to find the number of states that had a dropout rate less than 30% but had an increased dropout rate over the 2-year period. Since there were 12 states with an increased dropout rate, and we found that 6 of them had a dropout rate of at least 30%, it means there must be (12 - 6) states with increased dropout rates but less than a 30% dropout rate. This is the answer for part (b): 12 - 6 = 6 Thus, 6 states had a dropout rate less than 30% but had an increased dropout rate over the 2-year period. In summary: 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 had an increased dropout rate over the 2-year period.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Educational Data Analysis
Analyzing educational data involves collecting and interpreting information about students, schools, and educational systems to improve teaching, learning, and policy-making. Data can range from test scores and graduation rates to socioeconomic information and educational resources. Effective data analysis requires understanding the context of the data, recognizing patterns and trends, and turning findings into actionable insights.

For instance, in the presented exercise, data on high school dropout rates across various states is analyzed to understand the dynamics of education quality and challenges. Through organizing the data into understandable formats, like tables or charts, educators and policymakers can make well-informed decisions, such as implementing new educational programs targeted at states with higher dropout rates.
Inclusion-Exclusion Principle
The inclusion-exclusion principle is a fundamental concept in combinatorics and set theory used to count the elements in the union of two or more sets. This principle helps us to eliminate the overcounting that commonly occurs when sets intersect.

The formula shows how to calculate the size of the union of two sets, A and B, as the sum of the sizes of the individual sets minus the size of their intersection. In the exercise's context, the principle helps determine the number of states with both a high dropout rate (30% or higher) and an increase in that rate over the past two years by considering the union and intersection of these two groups.
High School Dropout Rates
The high school dropout rate is a critical indicator of educational system performance and student well-being. It refers to the percentage of students who leave high school before graduation without obtaining a diploma. Factors contributing to high dropout rates can include socioeconomic conditions, family issues, and lack of engagement with school.

Understanding dropout statistics can guide interventions aimed at preventing students from leaving school early. For example, a closer look at the states with increased dropout rates, as seen in the exercise, may reveal underlying issues that need addressing to improve retention and graduation rates.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In a survey of 200 employees of a company regarding their \(401(\mathrm{k})\) investments, the following data were obtained: 14 I had investments in stock funds. 91 had investments in bond funds. 60 had investments in money market funds. 47 had investments in stock funds and bond funds. 36 had investments in stock funds and money market funds. 36 had investments in bond funds and money market funds. 5 had investments only in some other vehicle. a. How many of the employees surveyed had investments in all three types of funds? b. How many of the employees had investments in stock funds only?

Let \(U=\\{1,2,3,4,5,6,7,8,9,10\\}\) \(A=\\{1,3,5,7,9\\}, B=\\{2,4,6,8,10\\}\), and \(C=\\{1,2,4\) \(5,8,9\\}\). List the elements of each set. a. \(A^{c} \cap\left(B \cap C^{c}\right)\) b. \(\left(A \cup B^{c}\right) \cup\left(B \cap C^{c}\right)\) c. \((A \cup B)^{c} \cap C^{c}\)

An exam consists of ten true-or-false questions. Assuming that every question is answered, in how many different ways can a student complete the exam? In how many ways can the exam be completed if a student can leave some questions unanswered because, say, a penalty is assessed for each incorrect answer?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $$ \text { If } A \subseteq B, \text { then } A \cap B=A \text { . } $$

Let \(A\) and \(B\) be subsets of a universal set \(U\) and suppose \(n(U)=200, n(A)=100, n(B)=80\), and \(n(A \cap B)=\) 40. Compute: a. \(n(A \cup B)\) b. \(n\left(A^{c}\right)\) c. \(n(A \cap B)\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.