91Ó°ÊÓ

In the Focus Problem at the beginning of this chapter, a study was described comparing the hatch ratios of wood duck nesting boxes. Group I nesting boxes were well separated from each other and well hidden by available brush. There were a total of 474 eggs in group I boxes, of which a field count showed about 270 hatched. Group II nesting boxes were placed in highly visible locations and grouped closely together. There were a total of 805 eggs in group II boxes, of which a field count showed about 270 hatched. (a) Find a point estimate \(\hat{p}_{1}\) for \(p_{1}\), the proportion of eggs that hatch in group I nest box placements. Find a \(95 \%\) confidence interval for \(p_{1}\). (b) Find a point estimate \(\hat{p}_{2}\) for \(p_{2}\), the proportion of eggs that hatch in group II nest box placements. Find a \(95 \%\) confidence interval for \(p_{2}\). (c) Find a \(95 \%\) confidence interval for \(p_{1}-p_{2} .\) Does the interval indicate that the proportion of eggs hatched from group I nest boxes is higher than, lower than, or equal to the proportion of eggs hatched from group II nest boxes? (d) What conclusions about placement of nest boxes can be drawn? In the article discussed in the Focus Problem, additional concerns are raised about the higher cost of placing and maintaining group I nest box placements. Also at issue is the cost efficiency per successful wood duck hatch.

Step by step solution

01

Calculate Point Estimate for Group I

To find \( \hat{p}_1 \), the point estimate of the proportion of eggs that hatched in Group I, divide the number of hatched eggs by the total eggs in that group.\[ \hat{p}_1 = \frac{270}{474} \approx 0.5696 \]

02

Compute 95% Confidence Interval for Group I

Using the point estimate \( \hat{p}_1 = 0.5696 \), the formula for the confidence interval is:\[ \hat{p}_1 \pm Z \sqrt{\frac{\hat{p}_1 (1 - \hat{p}_1)}{n_1}} \]Where \( Z \approx 1.96 \) for a 95% confidence level and \( n_1 = 474 \).Calculate the margin of error:\[ 1.96 \times \sqrt{\frac{0.5696 \times (1 - 0.5696)}{474}} \approx 0.0451 \]Thus, the confidence interval is:\[ (0.5696 - 0.0451, 0.5696 + 0.0451) = (0.5245, 0.6147) \]

03

Calculate Point Estimate for Group II

To find \( \hat{p}_2 \), divide the number of hatched eggs by the total eggs in Group II.\[ \hat{p}_2 = \frac{270}{805} \approx 0.3354 \]

04

Compute 95% Confidence Interval for Group II

Using \( \hat{p}_2 = 0.3354 \), apply the same formula for the confidence interval:\[ \hat{p}_2 \pm Z \sqrt{\frac{\hat{p}_2 (1 - \hat{p}_2)}{n_2}} \]With \( n_2 = 805 \), calculate the margin of error:\[ 1.96 \times \sqrt{\frac{0.3354 \times (1 - 0.3354)}{805}} \approx 0.0332 \]Thus, the confidence interval is:\[ (0.3354 - 0.0332, 0.3354 + 0.0332) = (0.3022, 0.3686) \]

05

Calculate 95% Confidence Interval for Difference

The formula for the confidence interval for \( p_1 - p_2 \) is:\[ (\hat{p}_1 - \hat{p}_2) \pm Z \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \]First, compute the difference:\[ \hat{p}_1 - \hat{p}_2 = 0.5696 - 0.3354 = 0.2342 \]Next, find the margin of error:\[ 1.96 \times \sqrt{\frac{0.5696(1-0.5696)}{474} + \frac{0.3354(1-0.3354)}{805}} \approx 0.0568 \]Thus, the confidence interval is:\[ (0.2342 - 0.0568, 0.2342 + 0.0568) = (0.1774, 0.2910) \]

06

Analyze Confidence Interval for Proportion Difference

As the confidence interval for \( p_1 - p_2 \) is entirely above 0, it suggests that the proportion of hatched eggs is higher in Group I nesting boxes than in Group II.

07

Conclusions and Recommendations

Group I boxes result in a higher hatch rate but may be less cost-efficient due to higher placement and maintenance costs. Group II boxes, while having lower hatch rates, are more visible and easier to maintain, which may lower costs.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Point Estimate

In the world of statistics, a point estimate is an invaluable tool for drawing insights from data. It is a single value used to approximate a population parameter. In this exercise, the parameter of interest is the proportion of eggs that hatched in different nest box groups. When we calculate the point estimate \( \hat{p}_1 = \frac{270}{474} \) for Group I and \( \hat{p}_2 = \frac{270}{805} \) for Group II, we are essentially estimating the true hatching proportion of all eggs in each group using the samples we observe.
Here, each point estimate gives a snapshot of the "best guess" of the population parameter based on the current data.
These estimates provide a starting point for more detailed statistical analyses, like calculating a confidence interval.

Proportion

Proportion is a fundamental concept in statistics, especially when working with categorical data. It represents the fraction of the whole that has a certain property.
In this context, it tells us what fraction of eggs in each group actually hatched. For example, the proportion \( p_1 \) in Group I is given by the fraction of hatched eggs to total eggs, which is \( \hat{p}_1 = 0.5696 \).
Similarly, for Group II, \( p_2 \) is \( \hat{p}_2 = 0.3354 \).
By expressing the number of successful outcomes (hatched eggs) over the total number of trials (total eggs), proportions help in easily comparing different groups and understanding the data distribution.

Margin of Error

The margin of error is what gives confidence intervals their range, allowing us to measure the uncertainty associated with point estimates. It tells us how much we can expect our point estimate to vary due to the randomness inherent in sampling.
When we compute a margin of error, calculated as \( 1.96 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), we account for the sample size and the variability of the parameter. For Group I, this results in a margin of error of approximately 0.0451, and for Group II, it is about 0.0332.
The margin of error is crucial in interpreting confidence intervals, because it shows the range within which the true population proportion is likely to lie, if the sampling were repeated.

Statistical Analysis

Statistical analysis involves a systematic approach to interpreting data and making decisions based on calculated probabilities and intervals. This exercise particularly highlights the application of confidence intervals to compare proportions from two groups.
By calculating a confidence interval for the difference between two proportions, we could assess if one proportion is statistically significantly different from another. In this example, the interval for \( p_1 - p_2 \) came out to be \([0.1774, 0.2910]\), entirely above zero.
This confidently suggests that Group I has a higher proportion of hatching eggs compared to Group II.
Such analysis not only helps in understanding data relationships but also aids in decision-making, such as evaluating the cost-effectiveness of different nesting strategies.