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Chapter 4: Problem 46

The Wall Street Journal/Harris Personal Finance poll asked 2082 adults if they owned a home (All Business website, January 23,2008 ). A total of 1249 survey respondents answered Yes. Of the 450 respondents in the \(18-34\) age group, 117 responded Yes. a. What is the probability that a respondent to the poll owned a home? b. What is the probability that a respondent in the \(18-34\) age group owned a home? c. What is the probability that a respondent to the poll did not own a home? d. What is the probability that a respondent in the \(18-34\) age group did not own a home?

Short Answer

Expert verified
a) 0.6, b) 0.26, c) 0.4, d) 0.74

Step by step solution

01

Understanding Probability Calculation

To calculate probability, we use the formula \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). The favorable outcome is the number of respondents who answered 'Yes', indicating they own a home.
02

Total Probability of Owning a Home

Calculate the probability that any respondent from the entire survey owns a home. There are 1249 respondents who answered 'Yes', and the total number of respondents is 2082. Thus, \( P(\text{own home}) = \frac{1249}{2082} \approx 0.6 \).
03

Probability of Owning a Home in Age Group 18-34

Calculate the probability for respondents in the 18-34 age group owning a home. There are 117 'Yes' answers out of 450 respondents in this age group. Thus, the probability is \( P(\text{own home | age 18-34}) = \frac{117}{450} \approx 0.26 \).
04

Total Probability of Not Owning a Home

To find the probability that a respondent does not own a home, subtract the probability of owning a home from 1: \( P(\text{not own home}) = 1 - P(\text{own home}) = 1 - 0.6 = 0.4 \).
05

Probability of Not Owning a Home in Age Group 18-34

For the age group 18-34, subtract the probability of owning a home from 1: \( P(\text{not own home | age 18-34}) = 1 - 0.26 = 0.74 \).

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Key Concepts

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

Understanding Statistics
Statistics plays an essential role in understanding data. It is the branch of mathematics dealing with data collection, organization, analysis, interpretation, and presentation. In our exercise, we are using statistics to analyze the outcome of a survey. Probability is a fundamental statistical concept here. It helps us to understand the likelihood of a specific outcome occurring. Probability is calculated using the basic formula: \[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \] This formula is crucial for determining how likely it is that a survey respondent owns a home, as seen in our example. By calculating probabilities, we understand the patterns and trends that our data represent. This understanding is used in various fields including finance, economics, and social sciences. It is fundamental for making predictions and informed decisions based on data.
Conducting a Survey Analysis
Survey analysis is a way to draw insights from data collected through questionnaires. In our exercise, 2082 adults were surveyed about home ownership. Surveys are powerful tools in gathering information about specific populations. When analyzing survey data, we need to:
  • Define the purpose of the survey
  • Identify the target population
  • Gather responses following a random or quota sampling
  • Analyze and interpret the collected data
Key to survey analysis is accurately calculating probabilities, which are reflective of the survey's results. For instance, in our survey, the probability that a respondent owns a home provides a general understanding of home ownership trends in the population studied. Survey analysis helps in understanding public opinions, behaviors, and characteristics. It supports strategic planning and policy making by providing actionable insights.
Age Group Analysis
Age group analysis involves breaking down data into segments based on different age categories. In the given survey data, the age group 18-34 is specifically examined.Analyzing specific age groups separately helps in identifying patterns or trends that might vary between age groups. For example, in our survey:
  • The probability that someone aged 18-34 owns a home is \( P(\text{own home | age 18-34}) = \frac{117}{450} \approx 0.26 \)
  • The probability that this age group does not own a home is \( P(\text{not own home | age 18-34}) = 0.74 \)
This shows that a lower proportion of this age group owns a home compared to the general population. Such analysis can be crucial for businesses and policymakers targeting programs or services towards specific demographics.Age group analysis helps in understanding how different segments of a population behave and is valuable in fields ranging from marketing to public health.

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Most popular questions from this chapter

Do you think the government protects investors adequately? This question was part of an online survey of investors under age 65 living in the United States and Great Britain (Financial Times/Harris Poll, October 1,2009 ). The number of investors from the United States and the number of investors from Great Britain who answered Yes, No, or Unsure to this question are provided as follows. a. Estimate the probability that an investor living in the United States thinks the government is not protecting investors adequately. b. Estimate the probability that an investor living in Great Britain thinks the government is not protecting investors adequately or is unsure the government is protecting investors adequately. c. For a randomly selected investor from these two countries, estimate the probability that the investor thinks the government is not protecting investors adequately. d. Based upon the survey results, does there appear to be much difference between the perceptions of investors living in the United States and investors living in Great Britain regarding the issue of the government protecting investors adequately?

Suppose that we have a sample space with five equally likely experimental outcomes: \(E_{1}\). \\[ \begin{aligned} E_{2}, E_{3}, E_{4}, E_{5} . \text { Let } \\ \qquad \begin{aligned} A &=\left\\{E_{1}, E_{2}\right\\} \\ B &=\left\\{E_{3}, E_{4}\right\\} \\ C &=\left\\{E_{2}, E_{3}, E_{5}\right\\} \end{aligned} \end{aligned} \\] a. \(\quad\) Find \(P(A), P(B),\) and \(P(C)\). b. Find \(P(A \cup B)\). Are \(A\) and \(B\) mutually exclusive? c. \(\quad\) Find \(A^{c}, C^{c}, P\left(A^{c}\right),\) and \(P\left(C^{c}\right)\). d. Find \(A \cup B^{c}\) and \(P\left(A \cup B^{c}\right)\). e. Find \(P(B \cup C)\).

A consulting firm submitted a bid for a large research project. The firm's management initially felt they had a \(50-50\) chance of getting the project. However, the agency to which the bid was submitted subsequently requested additional information on the bid. Past experience indicates that for \(75 \%\) of the successful bids and \(40 \%\) of the unsuccessful bids, the agency requested additional information. a. What is the prior probability of the bid being successful (that is, prior to the request for additional information)? b. What is the conditional probability of a request for additional information given that the bid will ultimately be successful? c. Compute the posterior probability that the bid will be successful given a request for additional information.

High school seniors with strong academic records apply to the nation's most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. Suppose that for a recent admissions class, an Ivy League college received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted \(18 \%\) of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was \(2375 .\) Let \(E, R,\) and \(D\) represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool. a. Use the data to estimate \(P(E), P(R),\) and \(P(D)\). b. Are events \(E\) and \(D\) mutually exclusive? Find \(P(E \cap D)\). c. For the 2375 students who were admitted, what is the probability that a randomly selected student was accepted during early admission? d. Suppose a student applies for early admission. What is the probability that the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

How many ways can three items be selected from a group of six items? Use the letters \(A, B\) \(\mathrm{C}, \mathrm{D}, \mathrm{E},\) and \(\mathrm{F}\) to identify the items, and list each of the different combinations of three items.
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