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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 \(51 \%\) of the respondents in the Cedar Rapids school district are in favor of the tax. The approval rating rises to \(56 \%\) for those with children in public schools. It falls to \(45 \%\) for those with no children in public schools. The older the respondent, the less favorable the view of the proposed tax: \(36 \%\) of those over age 56 said they would vote for the tax compared with \(72 \%\) 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. \(\quad P(F)\) ii. \(\quad P(F \mid C)\) iii. \(P\left(F \mid C^{C}\right)\) iv. \(P(F \mid O)\) v. \(\quad P(F \mid Y)\) b. Are \(F\) and \(C\) independent? Justify your answer. c. Are \(F\) and \(O\) independent? Justify your answer.

Step by step solution

01

Determine the Probabilities

Use the given percentage rates as the estimates for the probabilities. To convert a percentage to a probability, divide the percentage by 100.\nFor i. \(P(F) = \frac{51}{100}\) (probability that a voter favors the school tax),\nii. \(P(F|C) = \frac{56}{100}\) (probability that a voter with children in public schools favors the tax),\niii. \(P(F|C^C) = \frac{45}{100}\) (probability that a voter without children in public schools favors the tax),\niv. \(P(F|O) = \frac{36}{100}\) (probability that a voter aged over 56 favors the tax),\nv. \(P(F|Y) = \frac{72}{100}\) (probability that a voter aged 18-25 favors the tax).

02

Assess Independence of F and C

Two events are independent if the probability of one event is not affected by the occurrence of the other event. Stated formally, two events F and C are independent if and only if \(P(F) = P(F|C)\).\nCompute \(P(F) = P(F|C)\) to confirm whether they are equal. If equal, F and C are independent; if not, F and C are not independent.

03

Assess Independence of F and O

Following the same method as in Step 2, compute \(P(F) = P(F|O)\). If equal, F and O are independent; if not, 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.

Probability

Probability helps us understand the likelihood of an event taking place. In statistics, a probability is a number between 0 and 1 that indicates how likely an event is to occur. For example, when a voter poll in Cedar Rapids reports a 51% approval rating for a new school tax, this can be expressed as a probability, calculated by dividing the percentage by 100, giving us \( P(F) = \frac{51}{100} \).
This represents the probability that a randomly selected voter supports the school tax. Probabilities are foundational in statistics because they allow us to model real-world uncertainty quantitatively and make informed predictions.
Probabilities can change depending on certain conditions or parameters, such as the age of voters or whether they have children in public schools. Each provides a more nuanced interpretation of voter attitudes.

Independence

Two events are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other. This concept is vital because it helps clarify relationships between factors. For example, we are asked whether the event of supporting the tax \( F \) is independent of having children in public schools \( C \).
For events \( F \) and \( C \) to be independent, \( P(F) \) must equal \( P(F|C) \). If this condition holds, the percentage of voters favoring the tax would be the same regardless of whether they have children in public schools.
Using this understanding, we can analyze polling data to determine whether having children in public schools influences a voter's stance on the tax, or if other factors might be at play. If the probabilities differ, \( F \) and \( C \) are not independent.

Conditional Probability

Conditional probability is the probability that an event occurs, given that another event has already occurred. This concept is used to refine our probability estimates based on known information. For instance, the probability of supporting the school tax given that the voter has children in public schools is represented as \( P(F|C) \).
This probability is \( \frac{56}{100} \), meaning 56% of voters with children in public schools favor the tax. Conditional probability allows us to focus on particular segments of a population, giving us insights into how specific groups might differ in their views from the general populace.
Additionally, calculating conditional probabilities helps identify dependencies between events, offering a clearer understanding of factors influencing decisions, such as in voter polls where demographic factors might lead to different likelihoods of support for the tax.

Voter Poll Analysis

Analyzing voter polls involves interpreting the data collected to understand trends and implications. In this scenario, the poll helps us gauge voter support for a new school tax in Cedar Rapids by breaking down approval ratings among various demographic segments. For example, 72% of young voters aged 18-25 favor the tax, revealing a distinct trend among this demographic.
Using poll data, one can identify factors influencing public opinion, such as age or whether voters have children attending public schools. This analysis is crucial for lawmakers and policymakers seeking to understand community needs and priorities.
Poll analysis often involves computing probabilities and considering conditional probabilities to paint a more accurate picture. It's not just about understanding current opinions, but also about predicting how changes in policy or public information might sway voter support in the future.