91Ó°ÊÓ

Two thousand randomly selected adults were asked if they think they are financially better off than their parents. The following table gives the two- way classification of the responses based on the education levels of the persons included in the survey and whether they are financially better off, the same as. or worse off than their parents. $$\begin{array}{lccc} \hline & \begin{array}{c} \text { Less Than } \\ \text { High School } \end{array} & \begin{array}{c} \text { High } \\ \text { School } \end{array} & \begin{array}{c} \text { More Than } \\ \text { High School } \end{array} \\ \hline \text { Better off } & 140 & 450 & 420 \\ \text { Same as } & 60 & 250 & 110 \\ \text { Worse off } & 200 & 300 & 70 \end{array}$$ Suppose one adult is selected at random from these 2000 adults. Find the following probabilities. a. \(P(\) better off or high school \()\) b. \(P(\) more than high school or worse off \()\) c. \(P\) (better off or worse off)

Step by step solution

01

Calculate Total Number of Adults

The total number of adults in the survey is 2000. All the probabilities will be calculated based on this total number.

02

Probability of Better Off or High School

To get this, we add up the number of adults who are better off (140 + 450 + 420 = 1010) and the number of adults with high school education (450 + 250 + 300 = 1000). However, those who are both better off and have high school education (450) are included twice, so we subtract them. This gives a total of \(1010 + 1000 - 450 = 1560\). The probability is then \(\frac{1560}{2000} = 0.78\).

03

Probability of More than High School or Worse Off

To get this, we add up the number of adults with more than high school education (420 + 110 + 70 = 600) and those who are worse off (200 + 300 + 70 = 570). However, those who are both worse off and have more than high school education (70) are included twice, so we subtract them. This gives a total of \(600 + 570 - 70 = 1100\). The probability is then \(\frac{1100}{2000} = 0.55\).

04

Probability of Better Off or Worse Off

To get this, we add up the number of adults who are better off (1010) and those who are worse off (570). No one can be both better off and worse off, so no intersection to subtract. This gives a total of \(1010 + 570 = 1580\). The probability is then \(\frac{1580}{2000} = 0.79\).

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Key Concepts

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

Two-way classification

Two-way classification is a method used to organize data into a matrix form where categories intersect.
In this context, it allows for comparing two different variables: one along the rows, and one along the columns of the table.
For the survey, the rows represent the responses (better off, same as, and worse off) and columns depict different education levels (less than high school, high school, more than high school).
This type of classification helps in understanding which groups fall into particular categories, aiding in more nuanced analysis.

• **Rows**: Different financial perceptions.
• **Columns**: Levels of education attained.

With such structured data, it's simpler to apply statistical calculations for further analysis, like finding probabilities for different conditions.

Education levels

Education levels play a crucial role in survey data analysis by acting as a primary variable for comparison.
In this example, adults are categorized by their highest achieved education: less than high school, high school, and more than high school.
These categories help identify patterns in responses related to financial perceptions.

• **Less Than High School**: Individuals with minimal formal education.
• **High School**: Individuals who completed high school.
• **More Than High School**: Individuals who pursued education beyond high school.

Recognizing these differences is vital, as they can reveal how education affects people's perspectives about their financial situation compared to their parents.

Survey data analysis

Survey data analysis involves examining collected survey responses to draw meaningful conclusions.
This process leverages various statistical tools to interpret data - such as finding probabilities, trends, and correlations. By collating responses based on education levels and financial standing from the survey, analysts identify broader patterns, answering questions like:
1. Are certain education levels associated with feeling better or worse off financially?
2. Is there a common sentiment among people with the same educational qualifications? Survey analysis is crucial for forming insights that drive decisions and strategies. By understanding what influences individuals' financial perceptions, interventions can be tailor-made for improvement.

Statistical calculation

Statistical calculations are essential tools for analyzing survey data, enabling us to quantify trends and relationships.
In the provided example, simple probability calculations are performed to determine the likelihood of a randomly selected adult falling into specific categories. Here's a basic breakdown:

• **Addition Method**: When computing probabilities, if categories overlap (e.g., better off and high school), ensure to subtract the intersection to avoid double-counting.
• **Probability Formula**: The probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes).

For instance, calculating "better off or high school" involves addition and subtraction of overlapping individuals, ensuring we accurately assess this probability.
Such statistical calculations are fundamental in translating complex data into understandable results.