91Ó°ÊÓ

A Gallup poll conducted in 2017 asked people, "Do you think marijuana use should be legal?" In response, \(75 \%\) of Democrats, \(51 \%\) of Republicans, and \(67 \%\) of Independents said Yes. Assume that anyone who did not answer Yes answered No. Suppose the number of Democrats polled was 400 , the number of Republicans polled was 300 , and the number of Independents polled was 200 . a. Complete the two-way table with counts (not percentages). The first entry is done for you. Suppose a person is randomly selected from this group. b. What is the probability that the person is a Democrat who said Yes? c. What is the probability that the person is a Republican who said No? d. What is the probability that the person said No, given that the person is a Republican? e. What is the probahility that the person is a Republican given that the person said No? f. What is the probability that the person is a Democrat or a Republican?

Step by step solution

01

Translating Percentages to Actual Counts

First, calculate the number of people from each party responding Yes. For Democrats, it's \(75\%\) of 400, which equals 300. For Republicans, it's \(51\%\) of 300, which equals 153. For Independents, it's \(67\%\) of 200, which equals 134. Then, subtract these Yes responses from the total respondents for each party to get the number of No responses: For Democrats, it's 400-300=100, for Republicans it's 300-153=147, and for Independents it's 200-134=66.

02

Filling the Two-Way Table

With the information gained from step 1, fill in a two-way table. The rows represent the party affiliations (Democrats, Republicans, Independents) and the columns represent the response (Yes, No). Each cell contains the number of people from a particular party who gave a certain response.

03

Calculating the Probabilities

Now, calculate the total number of respondents, which is 400+300+200=900. For question b, the probability that the person is a Democrat who said Yes is 300 out of 900, i.e. \(\frac{300}{900}\). Similarly, calculate the probabilities for the remaining questions using the same approach.

04

Calculating Conditional Probabilities

For question d, find the probability that the person said No given that the person is a Republican. This is the number of Republicans who said No (147) divided by the total number of Republicans (300), i.e. \(\frac{147}{300}\). This is because the 'given' condition sets the total Republicans as the sample space.Likewise, calculate the probability that the person is a Republican given that the person said No. This is the number of No-saying Republicans (147) divided by the total number of No respondents (100 Democrats, 147 Republicans, 66 Independents), i.e. \(\frac{147}{100+147+66}\). This is because the 'given' condition sets the total No respondents as the sample space.

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Key Concepts

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

Two-way Table

A two-way table is a powerful tool for organizing data, especially when analyzing survey results. It allows us to see how different categories intersect and how responses are distributed among them. In the context of the Gallup poll question, a two-way table was used to show the counts of 'Yes' and 'No' responses from Democrats, Republicans, and Independents.

To create the table, we first needed to convert percentages into actual numbers, then fill each cell accordingly. For instance, if 75% of 400 Democrats answered 'Yes', the table cell for Democrats saying 'Yes' contains 300. Similarly, you calculate for others like Republicans and Independents.

• Two-way tables display data in a grid format, with rows representing one category (here, party affiliation) and columns representing another (response: Yes or No).
• By organizing the data this way, we can easily visualize and analyze trends and patterns.

Such tables simplify interpreting complex data, showing how different groups responded, and forming a basis for probability calculations.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. In analyzing poll data, knowing conditional probabilities can help understand specific group behaviors, such as how opinions on marijuana legalization differ within political affiliations.

To calculate this, you need to narrow down your sample space to the relevant subset. In our exercise, for example, if we want the probability someone said 'No' given they're a Republican, our sample space is only the Republicans.

• For Republicans who responded 'No', conditional probability is \(\frac{147}{300}\) because 147 out of 300 Republicans said 'No'.
• Conditional probability has its formula: \( P(A|B) = \frac{P(A \cap B)}{P(B)}\).

It enables a deeper understanding of specific scenarios and variations within the data. This is crucial for making informed inferences about population segments.

Poll Analysis

Poll analysis involves examining survey data to derive meanings and applicable insights. Through data analysis tools like two-way tables and statistical techniques like calculating probabilities, we can learn not just what people believe, but how these beliefs are distributed across different demographics.

By examining our poll, we determine the opinions of various party groups on the legality of marijuana. This sort of analysis helps identify trends like which party is more favorable towards or against the issue.

• A thorough poll analysis considers both the percentage and the actual count since percentages alone can be misleading without sample size context.
• It supports strategic decisions and policy-making by shedding light on public opinion in a structured, quantifiable manner.

Poll analysis provides insights into public opinion on key issues, helping society better understand differing viewpoints and possibly informing future legislation.

Party Affiliation

Party affiliation significantly influences opinion trends and survey responses. In the Gallup poll exercise, understanding how party membership correlates with perceptions of marijuana legalization is a key finding. Each political group—Democrats, Republicans, and Independents—demonstrates varied support levels.

Knowing these affiliations helps contextualize why poll results might lean a certain way. For instance, 75% of Democrats approved of legalization compared to 51% of Republicans.

• Party affiliation indicates the likely stance on various social and political issues.
• Analyzing these groups’ responses gives insight into potential policy support and opposition.

Understanding party affiliation offers a framework for predicting electoral support, shaping campaign strategies, and crafting public policies that resonate with constituents' views.