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Chapter 23: Problem 33

(Optional Topic) Lyme Disease. Lyme disease is spread in the northeastern United States by infected ticks. The ticks are infected mainly by feeding on mice, so more mice result in more infected ticks. The mouse population rises and falls with the abundance of acorns, their favored food. Experimenters studied two similar forest areas in a year when the acorn crop failed. They added hundreds of thousands of acorns to one area to imitate an abundant acorn crop, while leaving the other area untouched. The next spring, 54 of the 72 mice trapped in the first area were in breeding condition, versus 10 of the 17 mice trapped in the second area. 27 Estimate the difference between the proportions of mice ready to breed in good acorn years and bad acorn years. (Use \(90 \%\) confidence. Be sure to justify your choice of confidence interval.)

Short Answer

Expert verified
The difference in breeding proportions is approximately \(-0.0129\) to \(0.3365\) with 90% confidence.

Step by step solution

01

Identify the Proportions

First, identify the sample proportions of mice ready to breed in both areas. For the area with added acorns, 54 out of 72 mice were in breeding condition, so the proportion is \( \hat{p}_1 = \frac{54}{72} \). For the area without added acorns, 10 out of 17 mice were ready to breed, so the proportion is \( \hat{p}_2 = \frac{10}{17} \).
02

Calculate Sample Proportions

Calculate the actual sample proportions for each group. For area 1, \( \hat{p}_1 = \frac{54}{72} = 0.75 \). For area 2, \( \hat{p}_2 = \frac{10}{17} \approx 0.5882 \). These are the proportions of mice ready to breed in each area.
03

Find the Point Estimate of Difference

Calculate the point estimate of the difference between the two proportions \( \hat{p}_1 - \hat{p}_2 = 0.75 - 0.5882 = 0.1618 \). This is the estimated difference between the breeding proportions in the two areas.
04

Calculate the Standard Error

The standard error for the difference between two proportions is given by \[ SE = \sqrt{ \frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2} } \] where \( n_1 = 72 \) and \( n_2 = 17 \). Plug in the values: \[ SE = \sqrt{ \frac{0.75(1-0.75)}{72} + \frac{0.5882(1-0.5882)}{17} } \approx 0.1062 \]
05

Determine the Critical Value

For a 90% confidence interval, use a z-critical value. The critical value for a 90% confidence level is approximately 1.645.
06

Calculate the Confidence Interval

Use the formula for the confidence interval: \[ (\hat{p}_1 - \hat{p}_2) \pm Z \times SE \] Substitute the values: \[ 0.1618 \pm 1.645 \times 0.1062 \approx (0.1618 \pm 0.1747) \] This results in an interval of \(-0.0129, 0.3365\).
07

Interpret the Results

The 90% confidence interval for the difference in breeding proportions between areas with more acorns versus those with fewer is approximately \(-0.0129, 0.3365\). This suggests there is likely a difference but it could be as low as negative or as high as 33.65%.

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Key Concepts

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

Confidence Intervals
When we talk about confidence intervals, we are referring to a range of values within which we believe the true difference in breeding proportions likely lies. Confidence intervals offer a way to estimate this location with a given probability. In our exercise, we use a 90% confidence interval.

This means we can be 90% confident that the true difference between the two proportions of mice that are ready to breed (in areas with good versus poor acorn yields) falls within our calculated interval of \(-0.0129\) to \(0.3365\). This range not only gives us an estimate but quantifies our uncertainty about this estimate. Remember, a narrower interval indicates more precision, while a wider interval indicates more variability in our estimate.
  • The confidence interval is constructed around the point estimate.
  • The confidence level indicates how sure we are that the interval includes the true difference.
  • The interval is calculated using the critical value and the standard error.
Confidence intervals provide insight into the reliability and precision of our estimates derived from sample data.
Proportion Difference
Proportion difference is a core concept here, focusing on how different the proportions or percentages are between two groups. In this exercise, we calculated a difference in proportions of mice ready to breed in two different environments.

The first forest area had acorns added, resulting in a breeding proportion of \(\hat{p}_1 = 0.75\). In contrast, the second, untouched area had a proportion of \(\hat{p}_2 \approx 0.5882\). The point estimate of the difference, which is the basis for our further calculations, was calculated to be \(.75 - .5882 = 0.1618\).
  • The point estimate indicates the observed difference between the two groups.
  • It serves as the middle of the confidence interval.
  • This difference helps us understand the impact of an abundant acorn year on mice breeding behaviors.
Understanding the difference between proportions helps us discern the potential influence of variable conditions, such as food availability, on life processes like breeding.
Standard Error
The concept of standard error is crucial for understanding how much variability there is in our sample estimate of the proportion difference. The standard error gives us an idea of how precise our estimate is.

Mathematically, the standard error for the difference between two proportions is calculated using the formula \(SE = \sqrt{ \frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2} } \), where \(n_1\) and \(n_2\) are the sample sizes of the respective groups.

For our exercise, it was calculated that \(SE \approx 0.1062\), which is combined with the critical value to create the confidence interval.
  • The standard error indicates the reliability of the point estimate.
  • Smaller standard errors reflect more reliable estimates.
  • Larger sample sizes usually result in smaller standard errors.
A good grasp of standard error helps us assess the potential variation in our estimates and decide how much trust we can place in them.
Hypothesis Testing
Hypothesis testing is a statistical method used to make a decision about a population parameter based on sample data. In this context, we are interested in whether there is a significant difference in the breeding proportions of mice in the two areas.

In our scenario, the hypothesis might be: - **Null Hypothesis (H0):** There is no difference in the proportion of mice ready to breed between areas with and without added acorns.- **Alternative Hypothesis (H1):** There is a difference in these proportions.

To test this, we calculated a confidence interval. If this interval does not contain zero, it suggests that there is a significant difference, favoring the alternative hypothesis. However, in our case, since the interval does include zero (\(-0.0129\) to \(0.3365\)), we do not have strong enough evidence to reject the null hypothesis at a 90% confidence level.
  • Hypothesis testing helps decide if the observed difference is statistically significant.
  • The conclusion is based on whether the confidence interval includes zero.
  • It aids in making data-backed decisions about the effects of different conditions.
This method offers a structured approach to test ideas and make informed conclusions based on sample data observations.

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Most popular questions from this chapter

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