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Information from a poll of registered voters in Cedar Rapids, Iowa, to assess voter support for a new school tax was the basis for the following statements (Cedar Rapids Gazette, August 28,1999 ): The poll showed 51 percent of the respondents in the Cedar Rapids school district are in favor of the tax. The approval rating rises to 56 percent for those with children in public schools. It falls to 45 percent for those with no children in public schools. The older the respondent, the less favorable the view of the proposed tax: 36 percent of those over age 56 said they would vote for the tax compared with 72 percent of 18- to 25 -year-olds. Suppose that a registered voter from Cedar Rapids is selected at random, and define the following events: \(F=\) event that the selected individual favors the school \(\operatorname{tax}, C=\) event that the selected individual has children in the public schools, \(O=\) event that the selected individual is over 56 years old, and \(Y=\) event that the selected individual is \(18-25\) years old. a. Use the given information to estimate the values of the following probabilities: i. \(P(F)\) ii. \(P(F \mid C)\) iii. \(P\left(F \mid C^{C}\right)\) iv. \(P(F \mid O)\) v. \(P(F \mid Y)\) b. Are \(F\) and \(C\) independent? Justify your answer. c. Are \(F\) and \(O\) independent? Justify your answer.

Step by step solution

01

Identify the given information

The following information is provided: \n\n1. 51% of respondents favor the tax. \n2. The approval rating rises to 56% for those with children in public schools. \n3. The approval rating falls to 45% for those with no children in public schools. \n4. 36% of those over age 56 said they would vote for the tax. \n5. 72% of 18 - to 25 -year-olds said they would vote for the tax.

02

Calculate the individual probabilities

From the given information, we can directly determine: \n\ni. \(P(F)\) = 0.51 (probability that a randomly chosen individual supports the tax) \nii. \(P(F|C)\) = 0.56 (probability that a randomly chosen individual supports the tax given that they have children in public schools) \niii. \(P(F|C^C)\) = 0.45 (probability that an individual supports the tax given that they don't have children in public schools) \niv. \(P(F|O)\) = 0.36 (probability that a randomly chosen individual supports the tax given they are over the age of 56) \nv. \(P(F|Y)\) = 0.72 (probability that an individual supports the tax given they are between 18 and 25 years old)

03

Determine independence of F and C

Two events are independent if the occurrence of one does not change the probability of the other. Mathematically, two events A and B are independent if \(P(A|B) = P(A)\). So to determine if F and C are independent, we can compare \(P(F|C)\) to \(P(F)\). So if 0.56 (which is \(P(F|C)\)) equals 0.51 (which is \(P(F)\)), the two events are independent. Since these values are not equal, F and C are not independent.

04

Determine independence of F and O

Following the same reasoning as in the previous step, we compare \(P(F|O)\) to \(P(F)\). So if 0.36 (which is \(P(F|O)\)) equals 0.51 (which is \(P(F)\)), the two events are independent. Since these values are not equal, F and O are not independent.

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Key Concepts

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

Voter Polls

Voter polls are an essential tool in understanding public opinion on various issues. These surveys are typically conducted among a sample of registered voters to predict the outcome of votes or gauge support for specific measures, such as a new school tax like in Cedar Rapids, Iowa. Polls collect data by asking a series of questions about the respondents’ views. In this instance, half of the respondents (51%) favored the tax. Such information indicates a slight majority leaning towards favoring the new measure. Additionally, the data reveals interesting demographic trends, such as increased support (56%) among those with children in public schools. These insights can help stakeholders figure out what factors influence voter support. The intricate breakdown of responses further reflects different segments of the population. For instance: - 56% of respondents with children in public schools support the tax. - Support declines to 45% among those with no children in public schools. - Only 36% of voters over 56 support the tax. - A significant 72% of young voters aged 18-25 are in favor. The detailed segmentation highlights the importance of understanding specific voter groups when assessing the potential success of proposed measures.

Independence of Events

In probability and statistics, understanding the concept of independence is crucial, especially when determining the relationship between different events. Two events are considered independent if the occurrence of one does not affect the probability of the other. Mathematically, events A and B are independent if the probability of A given B is the same as the probability of A, or: \[ P(A|B) = P(A) \] Applying this to the problem of voter support for a new tax in Cedar Rapids, we look at whether having children in public schools (event C) affects the probability of supporting the tax (event F). The probability of favoring the tax generally is 0.51, while the probability given they have children in public schools is 0.56. Since 0.56 is different from 0.51, F and C are not independent – meaning having children in public schools increases the likelihood of supporting the new tax.Similarly, determining whether age (for instance, being over 56, event O) affects tax favorability, we compare \( P(F|O) = 0.36 \) to the overall probability \( P(F) = 0.51 \). Since these differ, F and O are not independent – with older age correlating with lower tax support.Recognizing dependencies among events helps in more accurate predictions and better insights into the factors influencing public opinion.

Statistics Problem Solving

Statistics problem-solving involves analyzing data and drawing conclusions based on probability theories. When engaged in exercises like evaluating voter polls, it's crucial to interpret information correctly and use methods systematically.Firstly, gather and understand the given information. For instance, you have percentages that indicate support levels across different demographics. These figures represent probabilities that help predict voting behaviors. Secondly, compute relevant probabilities as needed for the problem. By calculating initial probabilities like \( P(F) \), you establish a baseline understanding of support. Conditional probabilities, denoted as \( P(A|B) \), allow us to see how specific conditions (such as being a parent) affect these probabilities.Another critical step is checking event independence. Methods like comparing conditional probabilities to overall probabilities help us understand if certain factors independently influence others. Also, drawing on concepts of conditional probability aids in solving more complex statistical queries, as seen in the given scenario.Approaching statistics problems with a structured method will enhance your understanding and ability to apply probabilities in real-world contexts. Continuous practice with real life examples, like voter polls, sharpens statistical problem-solving skills.