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In early \(2003,\) President Bush proposed eliminating the taxation of dividends to shareholders on the grounds that it was double taxation. Corporations pay taxes on the earnings that are later paid out in dividends. In a poll of 671 Americans, TechnoMetrica Market Intelligence found that \(47 \%\) favored the proposal, \(44 \%\) opposed it, and \(9 \%\) were not sure (Investor's Business Daily, January 13,2003 ). In looking at the responses across party lines the poll showed that \(29 \%\) of Democrats were in favor, \(64 \%\) of Republicans were in favor, and \(48 \%\) of Independents were in favor. a. How many of those polled favored elimination of the tax on dividends? b. What is the conditional probability in favor of the proposal given the person polled is a Democrat? c. Is party affiliation independent of whether one is in favor of the proposal? d. If we assume people's responses were consistent with their own self- interest, which group do you believe will benefit most from passage of the proposal?

Step by step solution

01

Calculate total number in favor

To find the total number of respondents who favored eliminating the tax on dividends, use the percentage given: 47% of 671 people. This can be expressed as \( 0.47 \times 671 \). Calculate this value to find the number of people in favor.

02

Calculate the conditional probability for Democrats

The problem states that 29% of Democrats are in favor. Thus, given a person is a Democrat, the probability they support the proposal is 0.29. This is the conditional probability.

03

Determine Independence using Conditional Probability

To check independence between party affiliation and favoring the proposal, compare the overall probability of being in favor (0.47) with the conditional probabilities for each group. If the conditional probability of favoring does not equal 0.47 for any group, party affiliation affects the likelihood of favoring the proposal, indicating dependence.

04

Analyze benefits based on self-interest

Assuming self-interest motivation, the group with the highest percentage in favor likely benefits most. Here, 64% of Republicans are in favor, indicating they perceive the greatest benefit from the tax proposal.

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Key Concepts

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

Conditional Probability

Conditional probability helps us understand the likelihood of an event occurring when another event is already known to have happened. It is a crucial concept in business statistics, especially in analyzing survey results or any datasets. In the context of stakeholders, it allows us to deduce how a particular group feels about a proposal when we know their particular characteristic or affiliation.

For example, in the bush tax proposal, we learn that 29% of Democrats favored eliminating the tax on dividends. The conditional probability here represents the likelihood of being in favor given the condition that the individual is a Democrat. In mathematical terms, this is expressed as:

• \(P( ext{Favoring Proposal | Democrat}) = 0.29\)

This means that if we randomly select a Democrat from the survey, there's a 29% chance they support the proposal.

The calculation of conditional probabilities, like in this survey, provides deeper insights into specific segments of the population.

Independence in Probability

The concept of independence in probability is foundational for understanding the relationship between different variables. Two events are independent if the occurrence of one does not affect the probability of the other occurring. In the poll scenario regarding the tax proposal, this principle can be explored by comparing the overall probability of support against each party's probability.

If party affiliation was independent of the proposal support, the probability of favoring the proposal, regardless of party, would match the overall probability of support, which in this survey is 0.47. However, we see differing probabilities for each group:

• Democrats: 0.29
• Republicans: 0.64
• Independents: 0.48

Since these conditional probabilities are not equal to 0.47, we deduce that there is a dependence; party affiliation does impact attitudes towards the proposal. Independence assessments like this help in identifying biases or influential factors within data.

Survey Analysis

Survey analysis is a systematic way of studying survey data to extract meaningful insights. When analyzing surveys, it's essential to understand how the results can be segmented by demographic or categorical variables, like political affiliation in this example.

The Bush tax proposal survey queried citizens across different affiliations and yielded varied responses: 47% favored the proposal, 44% opposed it, and 9% were uncertain. Within these, the specific breakdowns reveal how each group aligns with or diverges from the overall sentiment.

• Represents opinions within sub-groups: For instance, only 29% of Democrats and 64% of Republicans showed support, highlighting the proposal's varying reception across political lines.
• Utilizes demographics to predict majority benefit: By assuming responses align with personal gain, one can infer that policies favoring dividends would predominantly benefit groups showing strong support.

Through careful survey analysis, businesses and policymakers can tailor strategies and predict which segments of the population will most support or oppose their initiatives, granting significant foresight into public reaction.