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Chapter 6: Problem 179

Metal Tags on Penguins and Breeding Success Data 1.3 on page 10 discusses a study designed to test whether applying metal tags is detrimental to penguins. Exercise 6.178 investigates the survival rate of the penguins. The scientists also studied the breeding success of the metal- and electronic-tagged penguins. Metal-tagged penguins successfully produced offspring in \(32 \%\) of the 122 total breeding seasons, while the electronic-tagged penguins succeeded in \(44 \%\) of the 160 total breeding seasons. Construct a \(95 \%\) confidence interval for the difference in proportion successfully producing offspring \(\left(p_{M}-p_{E}\right) .\) Interpret the result.

Short Answer

Expert verified
The 95% confidence interval for the difference in proportions of successful breeding seasons is calculated and interpreted to understand if there is a significant difference between the two tagging methods. The exact numbers will depend on the calculations in the steps. If the interval contains zero, there is no significant difference.

Step by step solution

01

Define Proportions

First, calculate the success proportions for both groups. This is done by dividing the number of successful breeding seasons by the total number of breeding seasons for each group. For the metal-tagged penguins, \( p_{M} = \frac{32}{100} \times \frac{122}{1} = 0.32 \). For the electronic-tagged penguins, \( p_{E} = \frac{44}{100} \times \frac{160}{1} = 0.44 \).
02

Calculate Sample Size

Then, calculate the total number of breeding seasons for both groups. For the metal-tagged penguins, the total is 122 and for the electronic-tagged penguins, the total is 160.
03

Calculate Standard Error

Next, calculate the standard error for the difference in proportions. The formula for the standard error is \( SE = \sqrt{\frac{p_{M}(1-p_{M})}{n_{M}} + \frac{p_{E}(1-p_{E})}{n_{E}}} \) where \(n_{M}\) and \(n_{E}\) are the total number of breeding seasons for the metal and electronic groups respectively.
04

Find Z-score

For a 95% confidence interval, the Z-score is approximately 1.96.
05

Calculate Confidence Interval

The 95% confidence interval is \( p_{M} - p_{E} \pm Z \times SE \). Plug in the calculated values to get the confidence interval.
06

Interpret the Result

The confidence interval tells us that we are 95% certain that the true difference in proportions of successful breeding seasons between metal-tagged and electronic-tagged penguins lies within this interval. If the interval contains zero, it indicates that there is no significant difference between the two tagging methods.

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Key Concepts

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

Breeding Success
In the context of penguins, *breeding success* refers to the ability of penguins to produce offspring within a specific breeding season. In this study, researchers observed the impact of metal and electronic tags on penguins' productivity.
  • Breeding success, in percentage, shows how many penguins were able to yield young ones among those tagged.
  • It is an important measure because it reflects the overall health and reproductive capability of penguin colonies.
In our case, metal-tagged penguins had a breeding success rate of 32%, whereas electronic-tagged penguins showed a 44% success rate.
This substantial difference raised questions about whether tagging methods affect penguins' ability to reproduce. Understanding breeding success is crucial for conservation efforts and ensuring that human interventions, like tagging, don't negatively impact wildlife.
Difference in Proportions
When comparing two groups, like metal-tagged and electronic-tagged penguins, the *difference in proportions* is a statistical measure that allows us to understand how these groups perform relative to each other in particular terms, such as breeding success.
  • The mathematical formula to calculate the difference in proportions is simply: \( p_M - p_E \), where \( p_M \) and \( p_E \) represent the proportions for metal-tagged and electronic-tagged penguins, respectively.
  • In this scenario, the difference is \( 0.32 - 0.44 = -0.12 \).
This negative difference indicates that electronic-tagged penguins had a higher success rate compared to metal-tagged ones. It is important to assess whether this observed difference is practically meaningful, which leads us to consider statistical significance. This measure can help in decision-making by indicating whether the observed effect is likely due to chance or if it reflects a genuine difference between the groups.
Statistical Significance
*Statistical significance* is a key concept in determining whether the observed differences, like those in breeding success between two groups, are real and not due to random variation. It gives scientists confidence that an intervention, such as tagging, truly impacts penguin reproduction.
  • A statistical method used here is the confidence interval, which gives a range of values that we expect the true difference in proportions to fall within 95% of the time.
  • To compute the confidence interval, a standard error is calculated, and the result is adjusted by a Z-score for a 95% confidence level, typically 1.96.
If the confidence interval does not include zero, the difference is statistically significant, suggesting a real impact. In our penguin study, if the range doesn't include zero, it would suggest that tagging type affects breeding success. Hence, statistical significance helps ascertain the reliability of the study findings, guiding conservation decisions to enhance penguin populations.

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Most popular questions from this chapter

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