91Ó°ÊÓ

A Pew Research poll asked 1,306 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies. $$\begin{array}{lcccc} &{\text { Response }} & \\ & \begin{array}{c}\text { Earth is } \\ \text { warming }\end{array} & \begin{array}{c}\text { Not } \\\\\text { warming } \end{array} & \begin{array}{c}\text { Don't Know } \\\\\text { Refuse }\end{array} & \text { Total } \\ \hline \text { Conservative Republican } & 0.11 & 0.20 & 0.02 & 0.33 \\ \text { Mod/Lib Republican } & 0.06 & 0.06 & 0.01 & 0.13 \\ \text { Mod/Cons Democrat } & 0.25 & 0.07 & 0.02 & 0.34 \\ \text { Liberal Democrat } & 0.18 & 0.01 & 0.01 & 0.20 \\ \hline \text { Total } & 0.60 & 0.34 & 0.06 & 1.00 \end{array}$$ (a) Are believing that the earth is warming and being a liberal Democrat mutually exclusive? (b) What is the probability that a randomly chosen respondent believes the earth is warming or is a liberal Democrat? (c) What is the probability that a randomly chosen respondent believes the earth is warming given that he is a liberal Democrat? (d) What is the probability that a randomly chosen respondent believes the earth is warming given that he is a conservative Republican? (e) Does it appear that whether or not a respondent believes the earth is warming is independent of their party and ideology? Explain your reasoning. (f) What is the probability that a randomly chosen respondent is a moderate/liberal Republican given that he does not believe that the earth is warming?

Step by step solution

01

Define Mutual Exclusivity

Two events are mutually exclusive if they cannot occur at the same time. Check if 'believing the earth is warming' and 'being a liberal Democrat' can occur simultaneously.

02

Verify if Events are Mutually Exclusive

From the table, 18% of respondents are liberal Democrats and believe the earth is warming. Since this percentage is not zero, these two events are not mutually exclusive.

03

Calculate Probability of Union of Events

To find the probability that a respondent believes the earth is warming or is a liberal Democrat, sum the probabilities of each event, then subtract the probability of both events occurring simultaneously: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), where \( P(A) = 0.60 \), \( P(B) = 0.20 \), and \( P(A \cap B) = 0.18 \).

04

Compute Probability Using Formula

Substitute and compute using the formula: \( P(A \cup B) = 0.60 + 0.20 - 0.18 = 0.62 \).

05

Probability Given Liberal Democrat

To find the probability that a respondent believes the earth is warming given that they are a liberal Democrat, use conditional probability: \( P(A|B) = P(A \cap B) / P(B) \), where \( P(A \cap B) = 0.18 \) and \( P(B) = 0.20 \).

06

Calculate Conditional Probability Using Formula

Substitute and compute: \( P(A|B) = 0.18 / 0.20 = 0.90 \).

07

Probability Given Conservative Republican

Calculate the probability that a conservative Republican believes the earth is warming as \( 0.11 \).

08

Test Independence

Two events are independent if \( P(A|B) = P(A) \). For belief in warming and each party/ideology group, check if conditional probabilities differ from overall probability \( P( \text{believes warming} ) = 0.60 \). For example, conservative Republicans show \( P(A|B)=0.11 \).

09

Conclusion on Independence

The probabilities of belief in warming under different party/ideology groups vary from the overall probability, indicating dependence between beliefs and party/ideology.

10

Probability Given Not Warming

To find the probability that a non-believing respondent is a moderate/liberal Republican, use \( P(B|\text{Not Warming}) = P(B \cap \text{Not Warming}) / P(\text{Not Warming}) \), where \( P(B \cap \text{Not Warming}) = 0.06 \) and \( P(\text{Not Warming}) = 0.34 \).

11

Calculate Final Conditional Probability

Substitute and compute: \( P(B|\text{Not Warming}) = 0.06 / 0.34 \approx 0.176 \).

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Key Concepts

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

Mutual Exclusivity

Understanding mutual exclusivity is important in probability theory. It helps us determine if two events can happen at the same time. Two events are mutually exclusive if they cannot occur simultaneously. If Event A occurs, then Event B cannot, and vice versa.

A good way to determine mutual exclusivity is by evaluating if the intersection of the two events is zero. In simpler terms, this means that there is zero probability that both events occur together. In our initial exercise, we consider events like 'believing the earth is warming' and 'being a liberal Democrat.'

If 18% of respondents are both liberal Democrats and believe the earth is warming, then these events are not mutually exclusive. There is overlap, meaning both can happen at the same time. Remember this principle when evaluating other scenarios too.

Conditional Probability

Conditional probability allows us to find the probability of an event occurring, given that another event has already occurred. Use the formula \(P(A|B) = \frac{P(A \cap B)}{P(B)}\) to calculate it. This means the probability of Event A given Event B equals the probability of both events occurring divided by the probability of Event B occurring.

For example, suppose we need to find the probability that a respondent believes the earth is warming given they are a liberal Democrat. Here, \(P(A \cap B)\) is 0.18 (both warming and being a liberal Democrat), and \(P(B)\) is 0.20 (just being a liberal Democrat). Plugging in these numbers, we find \(P(A|B) = \frac{0.18}{0.20} = 0.90\). This means there's a 90% probability that if someone is a liberal Democrat, they believe the earth is warming.

Conditional probability is useful in many fields, including marketing, where businesses predict customers' purchasing behavior based on certain conditions.

Statistical Independence

Statistical independence describes a scenario where the occurrence of one event does not affect the probability of another. Mathematically, two events A and B are independent if \(P(A|B) = P(A)\). This signifies that knowing Event B occurs does not change the probability of Event A.

In the context of the survey analysis task, we are asked if beliefs about global warming are independent of party ideology. Upon calculation, the conditional probabilities differ from the overall probability of believing the earth is warming. For instance, given a person is a conservative Republican, the probability they believe in warming is 0.11, which is different from the across-all-groups warming belief probability of 0.60.

This variation indicates dependency between one's belief about climate change and their political affiliation, showing that these beliefs are indeed not independent from party and ideology.

Survey Analysis

Survey analysis is a powerful tool in statistics used to extract meaningful insights from data collected. In the given problem, data is presented in terms of relative frequencies across different categories, such as political affiliations and ideologies.

To effectively analyze survey data, identify the key variables and how they interact. For instance, we examined how people's beliefs about warming relate to their political views. This required us to calculate probabilities and conditional probabilities to draw conclusions.

Survey analysis provides insights into the opinions and behaviors of different groups. It helps decision makers like policymakers and businesses understand public perception and tailor their strategies. Accurate survey analysis relies on understanding and interpreting statistical concepts, ensuring the correct application of probability theory.