91Ó°ÊÓ

Refer to the following setting. Yellowstone National Park surveyed a random sample of 1526 winter visitors to the park. They asked each person whether he or she owned, rented, or had never used a snowmobile. Respondents were also asked whether they belonged to an environmental organization (like the Sierra Club). The two-way table summarizes the survey responses. $$ \begin{array}{lcrr} \hline & {2}{c} {\text { Environmental Clubs }} & \\ { } & \text { No } & \text { Yes } & \text { Total } \\ \text { Never used } & 445 & 212 & 657 \\ \text { Snowmobile renter } & 497 & 77 & 574 \\ \text { Snowmobile owner } & 279 & 16 & 295 \\ \text { Total } & 1221 & 305 & 1526 \\ \hline \end{array} $$ Snowmobiles (5.2,5.3) (a) If we choose a survey respondent at random, what's the probability that this individual (i) is a snowmobile owner? (ii) belongs to an environmental organization or owns a snowmobile? (iii) has never used a snowmobile given that the person belongs to an environmental organization? (b) Are the events "is a snowmobile owner" and "belongs to an environmental organization" independent for the members of the sample? Justify your answer. (c) If we choose two survey respondents at random, what's the probability that (i) both are snowmobile owners? (ii) at least one of the two belongs to an environmental organization?

Step by step solution

01

Probability of Snowmobile Owner

To find the probability that a survey respondent is a snowmobile owner, divide the number of snowmobile owners by the total number of respondents. The number of snowmobile owners is 295, and the total number is 1526. Therefore, the probability is \( \frac{295}{1526} \approx 0.1933 \) or 19.33%.

02

Probability of Belonging to an Environmental Organization or Owning a Snowmobile

First, find the total number of people who either belong to an environmental organization or own a snowmobile by adding individuals who belog to an organization (305 total) and snowmobile owners (295 total). Then subtract those counted twice (both snowmobile owners and in an organization, 16). So, 305 + 295 - 16 = 584. Divide 584 by the total number of respondents, 1526. Thus, the probability is \( \frac{584}{1526} \approx 0.3827 \) or 38.27%.

03

Probability of Never Used Snowmobile Given Belongs to Environmental Organization

To find this conditional probability, divide the number of people who have never used a snowmobile and belong to an environmental organization (212) by the total number of people who belong to an environmental organization (305). Therefore, the probability is \( \frac{212}{305} \approx 0.6951 \) or 69.51%.

04

Check Independence of Being a Snowmobile Owner and Belonging to an Environmental Organization

Two events, A and B, are independent if \( P(A \cap B) = P(A) \times P(B) \). Here, \( P(A) \) is the probability of being a snowmobile owner \( \frac{295}{1526} \), \( P(B) \) is the probability of belonging to an environmental organization \( \frac{305}{1526} \), and \( P(A \cap B) \) is the probability of being both \( \frac{16}{1526} \). Calculate \( P(A) \times P(B) \approx 0.1933 \times 0.1999 \approx 0.0387 \) which is not equal to \( \frac{16}{1526} \approx 0.0105 \). Therefore, the events are not independent.

05

Probability Both Are Snowmobile Owners

The probability that the first respondent is a snowmobile owner is \( \frac{295}{1526} \). Assuming without replacement, the probability the second respondent is also a snowmobile owner is \( \frac{294}{1525} \). Thus, the probability both are owners is \( \frac{295}{1526} \times \frac{294}{1525} \approx 0.0372 \) or 3.72%.

06

Probability At Least One Belongs to an Environmental Organization

The probability that neither belongs to the environmental organization is the product of probabilities each does not belong: \( P(\text{not in org}) = \frac{1221}{1526} \) and for the second \( \frac{1220}{1525} \). Thus, \( \frac{1221}{1526} \times \frac{1220}{1525} \approx 0.6399 \). Therefore, the probability that at least one belongs is \( 1 - 0.6399 = 0.3601 \) or 36.01%.

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Key Concepts

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

Conditional Probability

Conditional probability is a way to find the likelihood of an event happening, given that another event has already happened. For example, we might be interested in knowing the probability that a person has never used a snowmobile, given they belong to an environmental organization. Here, the 'given' condition is crucial. We use the formula:

• \( P(A | B) = \frac{P(A \cap B)}{P(B)} \)

This formula tells us that the probability of event A happening, given that B has occurred, is equal to the probability of both A and B happening, divided by the probability of B.
In our problem, we find the number of visitors who never used a snowmobile and are part of an environmental group, then divide by those who are in such groups. This simplifies understanding situations where multiple conditions or known factors affect probabilities.

Two-Way Table

A two-way table is a useful organizing tool in probability and statistics. It shows how different variables relate to one another. Each cell in the table gives the count of individuals who share each combination of characteristics.

• Rows often represent one variable.
• Columns represent another variable.

For instance, our two-way table categorizes survey respondents by their snowmobile usage status and their membership in an environmental club. These tables help break down complex datasets into manageable information.
In practice, you can easily see how many people belong to each category by simply looking at the table. It's a visual approach that often makes interpreting and analyzing data much simpler. Two-way tables are especially helpful when calculating probabilities of combined events, like having one characteristic or belonging to a group.

Independence of Events

Independence of events is a key concept in probability. Two events are independent if the occurrence of one does not affect the occurrence of the other. Checking this relationship involves comparing probabilities:

• If \( P(A \cap B) = P(A) \times P(B) \), the events are independent.

Independence matters because it simplifies calculations. Suppose you have two events: being a snowmobile owner and belonging to an environmental organization. We calculated probabilities based on whether these were independent.
If they were independent, the probability of being both would equal the product of each event's individual probability. However, when they don't match, as in our example, it tells us these two characteristics interact in some significant way — they're not independent.

Random Sampling

Random sampling is a fundamental practice in statistics. It's used to gather a representative sample of a larger population. This method ensures that every individual has an equal chance of being selected, minimizing bias.

• It leads to more accurate and generalizable results.
• Random samples help statisticians make reliable inferences about the population as a whole.

In our scenario, survey respondents were randomly selected from winter visitors in the park. Thus, the data collected is likely to be unbiased and representative of the views of all park visitors in winter.
Random sampling helps ensure that the conclusions drawn from probability calculations like those in our exercise are valid. It forms a critical foundation for any statistical analysis, underpinning both its usefulness and reliability.