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Chapter 4: Problem 85

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. $$\begin{array}{lccc} \hline & \begin{array}{c} \text { Less Than } \\ \text { High School } \end{array} & \begin{array}{c} \text { High } \\ \text { School } \end{array} & \begin{array}{c} \text { More Than } \\ \text { High School } \end{array} \\ \hline \text { Better off } & 140 & 450 & 420 \\ \text { Same as } & 60 & 250 & 110 \\ \text { Worse off } & 200 & 300 & 70 \\ \hline \end{array}$$ a. Suppose one adult is selected at random from these 2000 adults. Find the following probabilities. i. \(P\) (better off and high school) ii. \(P(\) more than high school and worse off ) b. Find the joint probability of the events "worse off" and "better off." Is this probability zero? Explain why or why not.

Short Answer

Expert verified
i. The probability of a randomly selected adult being 'better off and high school' is 0.225. ii. The probability of a randomly selected adult being 'more than high school and worse off' is 0.035. The joint probability of 'worse off' and 'better off' is zero because these are contrasting states and cannot occur concurrently.

Step by step solution

01

Calculate probability of 'better off and high school'

The probability of a randomly selected adult being 'better off and high school' can be found by dividing the number of adults who are 'better off and high school' by the total number of adults. Using the data from the table, this calculation is \(\frac{450}{2000} = 0.225\)
02

Calculate probability of 'more than high school and worse off'

The probability of a randomly selected adult being 'more than high school and worse off' can be found in a similar way. Using the data from the table, this calculation is \(\frac{70}{2000} = 0.035\)
03

Explain joint probability of 'worse off' and 'better off'

The joint probability of 'worse off' and 'better off' would be zero because these are contrasting states and cannot occur at the same time. An individual cannot be both financially 'worse off' and 'better off' than their parents at the same time. Therefore, this question may need to be clarified

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Key Concepts

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

Joint Probability
Joint probability is a fundamental concept in statistics, often used to find the likelihood of two events occurring together. In the context of the exercise given, we are looking at occurrences such as "better off" and "high school," or "more than high school" and "worse off." These combinations represent specific scenarios of educational attainment and financial opinion compared to parents.

To calculate a joint probability, you start by identifying the count of people that satisfy both conditions. For example, with the combination "better off and high school," 450 respondents meet these criteria. Then, divide this number by the total population surveyed, which in this case is 2000. This gives a probability of \(P( ext{better off and high school}) = \frac{450}{2000} = 0.225\), or 22.5%.

Joint probabilities are always between 0 and 1. Zero indicates no overlap between the events, while 1 means complete overlap. In this exercise, there is a specific query about the joint occurrence of "worse off" and "better off," which is logically impossible because a person cannot be both. Hence, the joint probability is zero for these two states.
Two-way Classification Table
A two-way classification table is a useful statistical tool to display data arranged in rows and columns. It categorizes data by two different criteria, enabling us to see the distribution and relationships between two variables. In our exercise, the variables are educational level and financial status.

In the table provided, rows represent financial opinions ( "better off," "same as," "worse off"), while columns represent educational levels ("less than high school," "high school," "more than high school").
  • Each cell in the table contains the count of adults who fall into the combination of one financial opinion and one education level.

  • This layout quickly reveals patterns, such as whether higher education correlates with a better financial outlook.

  • It is a key component in calculating probabilities and conducting broader analyses.

This type of table is extensively used in statistics to summarize data and make findings more interpretable. It helps us understand how two variables may interact in a given dataset.
Education Level and Financial Status
The combination of education level and financial status provides meaningful insights into socioeconomic variables. In our dataset, education levels are divided into three categories: "Less than High School," "High School," and "More than High School." Financial status is recorded as opinions relative to the respondents' parents: "better off," "same as," and "worse off."

Analyzing the relationship between education and financial perception is crucial in understanding economic mobility and stability. Higher education levels often correlate with greater economic opportunities, which can translate to perceptions of being "better off."
  • Data indicates that more educated individuals tend to perceive themselves as financially better than their parents.
  • The interplay between education and financial perception can inform educational policies and economic strategies.
  • Understanding these trends can help address systemic inequalities.
Ultimately, this exercise underscores the importance of education in achieving financial well-being and improving societal conditions.

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Most popular questions from this chapter

A random sample of 250 adults was taken, and they were asked whether they prefer watching sports or opera on television. The following table gives the two-way classification of these adults $$\begin{array}{lcc} \hline & \begin{array}{c} \text { Prefer Watching } \\ \text { Sports } \end{array} & \begin{array}{c} \text { Prefer Watching } \\ \text { Opera } \end{array} \\ \hline \text { Male } & 96 & 24 \\ \text { Female } & 45 & 85 \end{array}$$ a. If one adult is selected at random from this group, find the probability that this adult i. prefers watching opera ii. is a male iii. prefers watching sports given that the adult is a female iv. is a male given that he prefers watching sports \(\mathbf{v}\). is a female and prefers watching opera vi. prefers watching sports or is a male b. Are the events "female" and "prefers watching sports" independent? Are they mutually exclusive? Explain why or why not.

A company that plans to hire one new employee has prepared a final list of six candidates, all of whom are equally qualified. Four of these six candidates are women. If the company decides to select at random one person out of these six candidates, what is the probability that this person will be a woman? What is the probability that this person will be a man? Do these two probabilities add up to \(1.0 ?\) If yes, why?

A small ice cream shop has 10 flavors of ice cream and 5 kinds of toppings for its sundaes. How many different selections of one flavor of ice cream and one kind of topping are possible?

The following table gives a two-way classification of all basketball players at a state university who began their college careers between 2001 and 2005, based on gender and whether or not they graduated \(\begin{array}{lcc} \hline & \text { Graduated } & \text { Did Not Graduate } \\ \hline \text { Male } & 126 & 55 \\ \text { Female } & 133 & 32 \\ \hline \end{array}\) a. If one of these players is selected at random, find the following probabilities. i. \(P(\) female and graduated \()\) ii. \(P(\) male and did not graduate \()\) b. Find \(P\) (graduated and did not graduate). Is this probability zero? If yes, why?

A certain state's auto license plates have three letters of the alphabet followed by a three-digit number. a. How many different license plates are possible if all three-letter sequences are permitted and any number from 000 to 999 is allowed? b. Arnold witnessed a hit-and-run accident. He knows that the first letter on the license plate of the offender's car was a \(\mathrm{B}\), that the second letter was an \(\mathrm{O}\) or a \(\mathrm{Q}\), and that the last number was a 5. How many of this state's license plates fit this description?
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