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The western United States has a number of four-lane interstate highways that cut through long tracts of wilderness. To prevent car accidents with wild animals, the highways are bordered on both sides with 12 -foot-high woven wire fences. Although the fences prevent accidents, they also disturb the winter migration pattern of many animals. To compensate for this disturbance, the highways have frequent wilderness underpasses designed for exclusive use by deer, elk, and other animals. In Colorado, there is a large group of deer that spend their summer months in a region on one side of a highway and survive the winter months in a lower region on the other side. To determine if the highway has disturbed deer migration to the winter feeding area, the following data were gathered on a random sample of 10 wilderness districts in the winter feeding area. Row \(B\) represents the average January deer count for a 5 -year period before the highway was built, and row \(A\) represents the average January deer count for a 5 -year period after the highway was built. The highway department claims that the January population has not changed. Test this claim against the claim that the January population has dropped. Use a \(5 \%\) level of significance. Units used in the table are hundreds of deer. \(\begin{array}{l|cccccccccc} \hline \text { Wilderness District } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\ \hline \text { B: Before highway } & 10.3 & 7.2 & 12.9 & 5.8 & 17.4 & 9.9 & 20.5 & 16.2 & 18.9 & 11.6 \\ \hline \text { A: After highway } & 9.1 & 8.4 & 10.0 & 4.1 & 4.0 & 7.1 & 15.2 & 8.3 & 12.2 & 7.3 \\ \hline \end{array}\)

Step by step solution

01

Define Hypotheses

Define the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\) for the test. The null hypothesis is \(H_0: \mu_B - \mu_A = 0\), which asserts that the average deer population before and after the highway construction remains the same. The alternative hypothesis is \(H_a: \mu_B - \mu_A > 0\), which suggests that the average deer population has decreased after the highway was built.

02

Calculate Differences

Calculate the difference between the before and after deer counts for each wilderness district. This will involve subtracting the after count from the before count. The differences are: 1.2, -1.2, 2.9, 1.7, 13.4, 2.8, 5.3, 7.9, 6.7, and 4.3.

03

Calculate Mean and Standard Deviation of Differences

Determine the mean and standard deviation of these differences. Sum the differences and divide by the number of districts to get the mean, which is 4.5. Use the formula \( s = \sqrt{\frac{\sum{(d_i - \bar{d})^2}}{n-1}} \) to find the standard deviation, which is approximately 4.284.

04

Calculate Test Statistic

Calculate the test statistic using the formula \( t = \frac{\bar{d} - 0}{s/\sqrt{n}} \) where \( \bar{d} \) is the mean difference, \( s \) is the standard deviation of the differences, and \( n \) is the number of pairs. The calculated \( t \) statistic is approximately 3.3178.

05

Determine Critical Value and Decision Rule

Find the critical value from the t-distribution table for a 95% confidence level (or 5% significance) with \( n-1 = 9 \) degrees of freedom. The critical t-value is 1.833. Since 3.3178 is greater than 1.833, we will reject the null hypothesis.

06

Conclusion

Since the test statistic exceeds the critical value, we reject the null hypothesis that the deer population remained unchanged. The data support the claim that the highway construction has led to a significant drop in the deer population.

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Key Concepts

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

Null Hypothesis

The null hypothesis, denoted as \( H_0 \), forms the foundation for statistical testing. It represents a statement or prediction that there is no effect or change in a given scenario. In this particular exercise involving deer populations, the null hypothesis holds the assertion that the average deer population before and after the highway was constructed has remained the same. Essentially, it states that the highway has not impacted the migration patterns or population numbers of the deer.

Formulating a null hypothesis is crucial because it provides a baseline to test against. If we can't reject the null hypothesis with the data we have, we might conclude that any observed effect is due to chance. In shorthand, the null hypothesis for this problem is written as \( H_0: \mu_B - \mu_A = 0 \), where \( \mu_B \) and \( \mu_A \) represent the mean deer count before and after the construction, respectively. This hypothesis is tested statistically to see if there's any evidence against it.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_a \), is a statement that contradicts the null hypothesis. It proposes that there is indeed an effect or a difference — in our case, it suggests a change in the deer population as an impact of the highway construction.

For this problem, the alternative hypothesis is \( H_a: \mu_B - \mu_A > 0 \). This indicates that the average deer population has decreased after the highway was built. The alternative hypothesis is usually what researchers expect or hope to find; it's essentially making a case against the null hypothesis.

Formulating an alternative hypothesis is how we assert that a significant change or relationship exists. It is crucial because it impacts our interpretation of the data. If we find enough statistical evidence, we may reject the null hypothesis in favor of the alternative.

Test Statistic

A test statistic arises from sample data and is used to decide whether or not to reject the null hypothesis. It's a numerical summary of the data that calculates how far the statistic stands from the null hypothesis in units of standard error. Put simply, the test statistic helps determine whether observed data deviate significantly from what would be expected under the null hypothesis.

In our scenario, we use the t-test since we're working with smaller samples. The formula, \( t = \frac{\bar{d} - 0}{s/\sqrt{n}} \), involves the mean of the differences \( \bar{d} \), the standard deviation \( s \), and the number of pairs \( n \). Plugging in the values gives us a test statistic of approximately 3.3178.

Interpreting the test statistic requires comparing it to a critical value from the t-distribution. If the magnitude of the test statistic exceeds the critical value, we gain statistical evidence to reject the null hypothesis. In this example, the test statistic surpasses the critical value of 1.833, supporting the alternative hypothesis that the deer population decreased.

Significance Level

The significance level, often denoted by \( \alpha \), is a threshold set by the researcher before testing begins. It determines the probability of rejecting the null hypothesis when it is true (type I error). In simpler terms, it represents how "confident" we need to be before concluding that a result is not due to random chance alone.

For this deer migration study, a 5% significance level is selected. This means we are allowing for a 5% risk of concluding that the deer population has decreased due to the highway when it has not.

With a 5% significance level, we are comparing our test statistic to a critical value from a t-distribution with 9 degrees of freedom (since there are 10 district pairs). The critical value is 1.833. For us to reject the null hypothesis, the test statistic needs to exceed this critical value. Since our calculated test statistic is approximately 3.3178, which is greater than the critical value, the decision falls toward rejecting the null hypothesis and accepting the alternative, suggesting a significant effect.