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Chapter 6: Problem 6

The article "Men, Women at Odds on Gun Control" (Cedar Rapids Gazette, September 8,1999 ) included the following statement: "The survey found that 56 percent of American adults favored stricter gun control laws. Sixtysix percent of women favored the tougher laws, compared with 45 percent of men." These figures are based on a large telephone survey conducted by Associated Press Polls. If an adult is selected at random, are the outcomes selected adult is female and selected adult favors stricter gun control independent or dependent outcomes? Explain.

Short Answer

Expert verified
The events 'selected adult is female' and 'selected adult favors stricter gun control' are dependent according to the given data.

Step by step solution

01

Calculate the probability of independent events

An important thing to note here is the definition of independent events. Two events A and B are independent if and only if the probability of A and B occurring together (in this case, a randomly picked adult being female and favoring stricter gun control), denoted as P(A and B), is equal to the product of the probabilities of A and B occurring separately, denoted as P(A)P(B). Using this, let's determine if both events occurring in this problem are independent or dependent.
02

Determine the probabilities of the individual events

From the provided data, we know that the probability of a randomly selected adult favoring stricter gun control, denoted as P(G), is 0.56 (or 56%) and the probability of a randomly selected adult being a woman, denoted as P(W), is 0.5 (or 50%, since there are an equal number of men and women in the total population).
03

Determine the probability of both events occurring together

We also know from the exercise that 66% of women favored stricter gun control, this can be denoted as P(G and W). Now that we have P(G), P(W), and P(G and W), we can check if the events are independent or dependent.
04

Compare the probabilities

If P(G and W) is equal to P(G)P(W), then the events are independent. If not, they are dependent. In this case, P(G and W) is 0.66, and P(G)P(W) is 0.56*0.5 = 0.28. As 0.66 is not equal to 0.28, the two events are dependent.

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Key Concepts

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

Probability Theory
At its core, probability theory is the mathematical framework used for analyzing uncertain events and quantifying the chance that certain outcomes will occur. The core concept lies in understanding the difference between independent and dependent events. Independent events have no impact on the likelihood of each other occurring, such as flipping a coin and rolling a die. The outcome of the coin flip does not influence the die roll. On the other hand, dependent events affect each other鈥檚 probabilities. In the context of the exercise, the event of an adult being female and the event that the adult favors stricter gun control laws are not independent because the probability of a woman favoring stricter gun laws is influenced by gender-specific opinions and trends.
Statistical Survey Analysis
Statistical survey analysis involves collecting, interpreting, and summarizing data from samples of a larger population. It's a critical tool for understanding opinions, behaviors, and patterns among groups of people, such as determining public opinion on gun control laws. Key to this process is obtaining a representative sample, ensuring accuracy in data collection, and properly analyzing the results to avoid bias. The Cedar Rapids Gazette exercise demonstrates how survey results can highlight relationships between different variables鈥攇ender and opinions on gun control鈥攊n a population. As surveys aim to reflect the larger population, it's essential to consider the probability of encountering a particular opinion or characteristic within the sample to make inferences accurately.
Gun Control Opinions Statistics
Gun control opinions statistics involve quantifying the attitudes and beliefs of individuals or groups regarding gun control measures. For instance, from the given exercise, we learn that 56 percent of American adults favored stricter gun laws, which is a crucial statistic that policymakers might use to understand public sentiment and inform legislation. When delving into this topic, it's important to break down the data further by demographics, as differing views based on gender are evident鈥攚ith a discrepancy between women (66%) and men (45%). These statistics not only provide insight into the current consensus but also can be employed to observe trends over time and assess the effectiveness of advocacy efforts on public opinion.

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Most popular questions from this chapter

USA Today (March 15,2001 ) introduced a measure of racial and ethnic diversity called the Diversity Index. The Diversity Index is supposed to approximate the probability that two randomly selected individuals are racially or ethnically different. The equation used to compute the Diversity Index after the 1990 census was $$\begin{aligned} 1-&\left[P(W)^{2}+P(B)^{2}+P(A I)^{2}+P(A P I)^{2}\right] \\ & \cdot\left[P(H)^{2}+P(\text { not } H)^{2}\right] \end{aligned}$$ where \(W\) is the event that a randomly selected individual is white, \(B\) is the event that a randomly selected individual is black, \(A I\) is the event that a randomly selected individual is American Indian, \(A P I\) is the event that a randomly selected individual is Asian or Pacific Islander, and \(H\) is the event that a randomly selected individual is Hispanic. The explanation of this index stated that 1\. \(\left[P(W)^{2}+P(B)^{2}+P(A I)^{2}+P(A P I)^{2}\right]\) is the probability that two randomly selected individuals are the same race 2\. \(\left[P(H)^{2}+P(\text { not } H)^{2}\right]\) is the probability that two randomly selected individuals are either both Hispanic or both not Hispanic 3\. The calculation of the Diversity Index treats Hispanic ethnicity as if it were independent of race a. What additional assumption about race must be made to justify use of the addition rule in the computation of \(\left[P(W)^{2}+P(B)^{2}+P(A I)^{2}+P(A P I)^{2}\right]\) as the probability that two randomly selected individuals are of the same race? b. Three different probability rules are used in the calculation of the Diversity Index: the Complement Rule, the Addition Rule, and the Multiplication Rule. Describe the way in which each is used.

Researchers at UCLA were interested in whether working mothers were more likely to suffer workplace injuries than women without children. They studied 1400 working women, and a summary of their findings was reported in the San Luis Obispo Telegram-Tribune (February 28,1995 ). The information in the following table is consistent with summary values reported in the article: $$\begin{array}{l|cccc} & & & \text { Children, } & \\ & \text { No } & \text { Children } & \text { but None } & \\ & \text { Children } & \text { Under 6 } & \text { Under 6 } & \text { Total } \\\ \hline \text { Injured on } & & & & \\ \text { the Job in } & & & & \\ 1989 & 32 & 68 & 56 & \mathbf{1 5 6} \\ \text { Not Injured } & & & & \\ \text { on the Job } & & & & \\ \text { in 1989 } & 368 & 232 & 644 & \mathbf{1 2 4 4} \\ \text { Total } & \mathbf{4 0 0} & \mathbf{3 0 0} & \mathbf{7 0 0} & \mathbf{1 4 0 0} \end{array}$$ The researchers drew the following conclusion: Women with children younger than age 6 are much more likely to be injured on the job than childless women or mothers with older children. Provide a justification for the researchers' conclusion. Use the information in the table to calculate estimates of any probabilities that are relevant to your justification.

Four students must work together on a group project. They decide that each will take responsibility for a particular part of the project, as follows: $$\begin{array}{lllll} \hline \text { Person } & \text { Maria } & \text { Alex } & \text { Juan } & \text { Jacob } \\ \text { Task } & \begin{array}{l} \text { Survey } \\ \text { design } \end{array} & \begin{array}{l} \text { Data } \\ \text { collection } \end{array} & \text { Analysis } & \begin{array}{l} \text { Report } \\ \text { writing } \end{array} \end{array}$$ Because of the way the tasks have been divided, one student must finish before the next student can begin work. To ensure that the project is completed on time, a timeline is established, with a deadline for each team member. If any one of the team members is late, the timely completion of the project is jeopardized. Assume the following probabilities: 1\. The probability that Maria completes her part on time is 8 . 2\. If Maria completes her part on time, the probability that Alex completes on time is \(.9\), but if Maria is late, the probability that Alex completes on time is only .6. 3\. If Alex completes his part on time, the probability that Juan completes on time is \(.8\), but if Alex is late, the probability that Juan completes on time is only \(.5\). 4\. If Juan completes his part on time, the probability that Jacob completes on time is \(.9\), but if Juan is late, the probability that Jacob completes on time is only .7. Use simulation (with at least 20 trials) to estimate the probability that the project is completed on time. Think carefully about this one. For example, you might use a random digit to represent each part of the project (four in all). For the first digit (Maria's part), \(1-8\) could represent on time and 9 and 0 could represent late. Depending on what happened with Maria (late or on time), you would then look at the digit representing Alex's part. If Maria was on time, \(1-9\) would represent on time for Alex, but if Maria was late, only 1-6 would represent on time. The parts for Juan and Jacob could be handled similarly.

A radio station that plays classical music has a "by request" program each Saturday evening. The percentages of requests for composers on a particular night are as follows: \(\begin{array}{lr}\text { Bach } & 5 \% \\ \text { Mozart } & 21 \% \\ \text { Beethoven } & 26 \% \\ \text { Schubert } & 12 \% \\ \text { Brahms } & 9 \% \\ \text { Schumann } & 7 \% \\ \text { Dvorak } & 2 \% \\ \text { Tchaikovsky } & 14 \% \\ \text { Mendelssohn } & 3 \% \\ \text { Wagner } & 1 \%\end{array}\) Suppose that one of these requests is randomly selected. a. What is the probability that the request is for one of the three B's? b. What is the probability that the request is not for one of the two S's? c. Neither Bach nor Wagner wrote any symphonies. What is the probability that the request is for a composer who wrote at least one symphony?

In a small city, approximately \(15 \%\) of those eligible are called for jury duty in any one calendar year. People are selected for jury duty at random from those eligible, and the same individual cannot be called more than once in the same year. What is the probability that an eligible person in this city is selected 2 years in a row? 3 years in a row?
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