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Chapter 7: Problem 11

In a survey of 2000 adults 18 yr and older conducted in 2007 , the following question was asked: Is your family income keeping pace with the cost of living? The results of the survey follow: $$\begin{array}{lcccc} \hline & \begin{array}{c} \text { Falling } \\ \text { behind } \end{array} & \begin{array}{c} \text { Staying } \\ \text { even } \end{array} & \begin{array}{c} \text { Increasing } \\ \text { faster } \end{array} & \begin{array}{c} \text { Don't } \\ \text { know } \end{array} \\ \hline \text { Respondents } & 800 & 880 & 240 & 80 \\ \hline \end{array}$$ Determine the empirical probability distribution associated with these data.

Short Answer

Expert verified
The empirical probability distribution associated with the survey data is: \( P(\text{Falling Behind}) = 0.4 \) \( P(\text{Staying Even}) = 0.44 \) \( P(\text{Increasing Faster}) = 0.12 \) \( P(\text{Don't Know}) = 0.04 \)

Step by step solution

01

Calculate the Total Number of Respondents

Add up the number of respondents in each category to calculate the total number of respondents in the survey. Total respondents = 800 (Falling Behind) + 880 (Staying Even) + 240 (Increasing Faster) + 80 (Don't Know) = 2000
02

Calculate the Empirical Probabilities for each Category

Divide the number of respondents in each category by the total number of respondents. a) Empirical probability for Falling Behind: P(Falling Behind) = \(\frac{800}{2000} = 0.4 \) b) Empirical probability for Staying Even: P(Staying Even) = \(\frac{880}{2000} = 0.44 \) c) Empirical probability for Increasing Faster: P(Increasing Faster) = \(\frac{240}{2000} = 0.12 \) d) Empirical probability for Don't Know: P(Don't Know) = \(\frac{80}{2000} = 0.04 \)
03

Summarize the Empirical Probability Distribution

Present the calculated empirical probabilities in a table format. The empirical probability distribution associated with the survey data is as follows: $$\begin{array}{l|c} \hline \text{Category} & \text{Empirical Probability} \\ \hline \text{Falling Behind} & 0.4 \\ \text{Staying Even} & 0.44 \\ \text{Increasing Faster} & 0.12 \\ \text{Don't Know} & 0.04 \\ \hline \end{array}$$

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Key Concepts

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

Empirical Probability
Empirical probability is an approach to estimating probabilities that is based on observed data rather than theoretical models. It is especially useful in real-world situations where observing outcomes can provide insights into probable future occurrences. In our example, we have a survey with results given across four different categories.
Here is how you calculate empirical probability:
  • Identify the number of times an outcome occurs.
  • Divide this by the total number of observations.
For example, in the survey regarding family income, 800 out of 2000 respondents felt they were falling behind. Thus, the empirical probability of this category is calculated as 800 divided by 2000, which results in 0.4. This number represents a 40% chance based on the survey results.
Survey Analysis
Survey analysis is crucial for understanding data collected from questionnaires like the one described. It involves exploring the data, calculating probabilities, and interpreting the outcomes to glean insights. In our survey, data was collected on how people perceive their income relative to the cost of living. This data was grouped into four categories: 'Falling Behind,' 'Staying Even,' 'Increasing Faster,' and 'Don't Know'. By analyzing the distribution of responses, we gain a better understanding of the economic sentiment among the respondents.
Key aspects to consider in survey analysis:
  • Reliability of the data - ensure that the sample size (2000 respondents) and survey method were appropriate.
  • Results interpretation - linking data with real-world implications can offer meaningful insights into economic trends.
  • Comparison with other data sources to detect patterns or discrepancies in public perception or economic conditions.
Statistics Education
Statistics education is vital for helping people collect, analyze, and interpret data effectively. It equips individuals with the necessary skills to make informed decisions based on data, fostering a better understanding of various phenomena.
In this context, this survey exercise serves as an excellent practical example for students to apply statistical concepts. Students learn to handle real data, calculate empirical probabilities, and understand how distribution can reflect larger trends. Some educational points to focus on include:
  • The importance of empirical data - shows the variability and randomness present in real-world data.
  • Graphical representations - using graphs can simplify the visual comparison of data categories.
  • Validation of results - comparing calculated probabilities with prior studies or predictions to assess accuracy and reliability.
Data Interpretation
Data interpretation transforms raw data into meaningful conclusions. It involves analyzing probabilities, patterns, and trends to provide insightful answers to questions posed by surveys like the income question.
In the exercise, data interpretation is used to discern how people's perceptions of income growth relate to the economy's status. By interpreting the empirical probabilities, we determine which segments feel most financially pressured and which seem economically stable. Important steps in data interpretation:
  • Contextualizing data - understanding the background of the data collection period (2007, in this case) aids in understanding the responses.
  • Identifying outliers or unusual trends - understanding why a small or large percentage chose 'Don't Know' could indicate indecision or lack of information.
  • Converting numbers into action - knowing empirical probabilities can guide policy decisions or further research directives in economic planning.

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Most popular questions from this chapter

The following table, compiled in 2004, gives the percentage of music downloaded from the United States and other countries by U.S. users: $$ \begin{array}{lcccccccc} \hline \text { Country } & \text { U.S. } & \text { Germany } & \text { Canada } & \text { Italy } & \text { U.K. } & \text { France } & \text { Japan } & \text { Other } \\ \hline \text { Percent } & 45.1 & 16.5 & 6.9 & 6.1 & 4.2 & 3.8 & 2.5 & 14.9 \\\ \hline \end{array} $$ a. Verify that the table does give a probability distribution for the experiment. b. What is the probability that a user who downloads music, selected at random, obtained it from either the United States or Canada? c. What is the probability that a U.S. user who downloads music, selected at random, does not obtain it from Italy, the United Kingdom (U.K.), or France?

Data compiled by the Highway Patrol Department regarding the use of seat belts by drivers in a certain area after the passage of a compulsory seat-belt law are shown in the accompanying table. $$\begin{array}{lcc} \hline & & \text { Percent of } \\ \text { Drivers } & \begin{array}{c} \text { Percent } \\ \text { of Drivers } \\ \text { in Group } \end{array} & \begin{array}{c} \text { Group Stopped } \\ \text { for Moving } \end{array} \\ \hline \text { Group I } & \text { Violation } \\ \text { (using seat belts) } & 64 & .2 \\ \hline \text { Group II } & & \\ \text { (not using seat belts) } & 36 & .5 \\ \hline \end{array}$$ If a driver in that area is stopped for a moving violation, what is the probability that he or she a. Will have a seat belt on? b. Will not have a seat belt on?

A poll was conducted among 250 residents of a certain city regarding tougher gun-control laws. The results of the poll are shown in the table: $$ \begin{array}{lccccc} \hline & \begin{array}{c} \text { Own } \\ \text { Only a } \\ \text { Handgun } \end{array} & \begin{array}{c} \text { Own } \\ \text { Only a } \\ \text { Rifle } \end{array} & \begin{array}{c} \text { Own a } \\ \text { Handgun } \\ \text { and a Rifle } \end{array} & \begin{array}{c} \text { Own } \\ \text { Neither } \end{array} & \text { Total } \\ \hline \text { Favor } & & & & & \\ \text { Tougher Laws } & 0 & 12 & 0 & 138 & 150 \\ \hline \begin{array}{l} \text { Oppose } \\ \text { Tougher Laws } \end{array} & 58 & 5 & 25 & 0 & 88 \\ \hline \text { No } & & & & & \\ \text { Opinion } & 0 & 0 & 0 & 12 & 12 \\ \hline \text { Total } & 58 & 17 & 25 & 150 & 250 \\ \hline \end{array} $$ If one of the participants in this poll is selected at random, what is the probability that he or she a. Favors tougher gun-control laws? b. Owns a handgun? c. Owns a handgun but not a rifle? d. Favors tougher gun-control laws and does not own a handgun?

CAR THEFT Figures obtained from a city's police department seem to indicate that, of all motor vehicles reported as stolen, \(64 \%\) were stolen by professionals whereas \(36 \%\) were stolen by amateurs (primarily for joy rides). Of those vehicles presumed stolen by professionals, \(24 \%\) were recovered within \(48 \mathrm{hr}, 16 \%\) were recovered after \(48 \mathrm{hr}\), and \(60 \%\) were never recovered. Of those vehicles presumed stolen by amateurs, \(38 \%\) were recovered within \(48 \mathrm{hr}, 58 \%\) were recovered after \(48 \mathrm{hr}\), and \(4 \%\) were never recovered. a. Draw a tree diagram representing these data. b. What is the probability that a vehicle stolen by a professional in this city will be recovered within \(48 \mathrm{hr}\) ? c. What is the probability that a vehicle stolen in this city will never be recovered?

Refer to the following experiment: Urn A contains four white and six black balls. Urn B contains three white and five black balls. A ball is drawn from urn A and then transferred to urn B. A ball is then drawn from urn B. What is the probability that the transferred ball was white given that the second ball drawn was white?
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