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Usage of the Olympic National Park. U.S. National Parks that contain designated wilderness areas are required by law to develop and maintain a wilderness stewardship plan. The Olympic National Park, containing some of the most biologically diverse wilderness in the United States, had a survey conducted in 2012 to collect information relevant to the development of such a plan. National Park Service staff visited 30 wilderness trailheads in moderate-to high-use areas over a 60-day period and asked visitors as they completed their hike to complete a questionnaire. The 1019 completed questionaires, giving a response rate of \(50.4 \%\), provided each subject's opinions on the use and management of wilderness. In particular, there were 694 day users and 325 overnight users in the sample. a. Why do you think the National Park staff only visited trailheads in moderate-to high-use areas to obtain the sample? b. Assuming that the 1019 subjects represent a random sample of users of the wilderness areas in Olympic National Park, give a \(90 \%\) confidence interval for the proportion of day users. c. The response rate was \(49 \%\) for day users and \(52 \%\) for overnight users. Does this lessen any concerns you might have regarding the effect of nonresponse on the interval you obtained in part (b)? Explain briefly. d. Do you think it would be better to refer to the interval in part (b) as a confidence interval for the proportion of day users or the proportion of day users on the most popular trails in the park? Explain briefly.

Step by step solution

01

Understanding Sampling Location Selection

The National Park staff visited trailheads in moderate-to high-use areas because these areas are more indicative of general park traffic. High-use areas provide a larger and more diverse sample of visitors, which is important for gaining insights that are representative of the user base. Furthermore, focusing on these areas ensures a higher likelihood of obtaining responses from a more significant number of park visitors within a limited timeframe.

02

Calculating the Confidence Interval

To find a 90% confidence interval for the proportion of day users, we need to use the formula for a confidence interval for a proportion: \[ CI = \hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]Where \( \hat{p} \) is the sample proportion, \( z \) is the critical value from the standard normal distribution for a 90% confidence level (1.645), and \( n \) is the sample size.- Given \( \hat{p} = \frac{694}{1019} \approx 0.681 \) - \[ SE = \sqrt{\frac{(0.681)(1-0.681)}{1019}} \approx 0.0148 \]- Confidence Interval: \[ 0.681 \pm 1.645 \times 0.0148 = [0.656, 0.706] \]

03

Analyzing Nonresponse Bias

The response rate was 49% for day users and 52% for overnight users, which indicates nearly equivalent response rates between the two groups. This lessens concerns about the effect of nonresponse bias because the similar response rates suggest that the views of nonrespondents are likely not systematically different between these two types of users. As such, the sample can be considered more representative of the overall user base.

04

Evaluating the Confidence Interval Context

The confidence interval calculated in Step 2 should be referred to as a confidence interval for the proportion of day users on the most popular trails, rather than the entire park. This is because the sample was drawn specifically from moderate-to high-use areas, which are the most popular trails. Therefore, the interval reflects usage among this subgroup, not all potential park users.

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Key Concepts

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

Sampling Methods

Sampling methods refer to the techniques used to select a portion of a population for study. The goal is to gather data that accurately represents the larger group. In the case of Olympic National Park, the staff chose to focus on trailheads in moderate-to high-use areas. Why is this effective?

• These areas provide access to more visitors, making it easier to gather data efficiently over a limited period.
• Highly visited areas are more likely to yield insights into the general behavior and opinions of park users.

Focusing on these areas helps ensure that the sample is diverse, providing data that can inform effective management decisions for the park. It's important to choose sampling locations wisely to ensure that the findings can be generalized to the population of interest. This approach helps balance between efficient data collection and improving the representativeness of the sample.

Confidence Interval

A confidence interval (CI) is a range of values that estimates a population parameter, based on sample data. It provides an indication of the precision of the estimate. In this particular scenario, we're interested in estimating the proportion of day users at the park with a 90% confidence level.To calculate the CI, we use the formula: \[ CI = \hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] Here,

• \( \hat{p} \) is the sample proportion, calculated by dividing the number of day users (694) by the total respondents (1019), approximately 0.681.
• \( z \) is the critical value from the standard normal distribution, which is 1.645 for a 90% confidence level.
• \( n \) is the sample size.

Using these values, we find the confidence interval for the proportion of day users to be between 0.656 and 0.706.This interval tells us that we can be 90% confident that the true proportion of day users in the sampled areas lies within this range.

Proportion

Proportion pertains to the part or fraction of the total sample that possesses a particular attribute. In this example, the proportion of interest is the ratio of day users to the total number of survey participants.When examining proportions, it is critical to understand:

• What the proportion represents: In this case, the proportion (\( \hat{p} = 0.681 \)) signifies the fraction of visitors to these trailheads who are day users.
• Why it's relevant: Tracking proportions help manage resources efficiently, such as staffing and facilities planning at frequent day-use areas.
• How it links to other metrics: Proportions can inform on trends, such as changing park usage patterns over time, which can guide policy-making and park management strategies.

Proportions provide a straightforward way to understand complex data about park visitors, enabling better decision-making to support park stewardship.

Nonresponse Bias

Nonresponse bias occurs when the responses collected in a survey are systematically different from the intended sample population. This can lead to skewed results if non-respondents differ significantly from respondents in terms of key characteristics or preferences. In this case, response rates of 49% for day users and 52% for overnight users suggest a somewhat balanced participation between these groups. Key considerations regarding nonresponse bias include:

• Assessing response rates: Similar response rates between key groups lessen concerns, as there's less likelihood of systematic difference affecting results.
• Impact on data: If nonrespondents have similar characteristics as respondents, bias is minimized, enhancing the reliability of the data collected.
• Mitigation strategies: Increase incentives, follow-up for nonrespondents, or use statistical adjustments to correct any potential bias.

While a perfect response rate is rare, understanding and mitigating nonresponse bias helps ensure that survey findings are as representative and accurate as possible.