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In environmental studies, sex ratios are of great importance. Wolf society, packs, and ecology have been studied extensively at different locations in the U.S. and foreign countries. Sex ratios for eight study sites in northern Europe are shown below (based on The Wolf by L. D. Mech, University of Minnesota Press). \(\begin{array}{lcc} \hline \text { Location of Wolf Pack } & \text { \% Males (Winter) } & \text { \% Males (Summer) } \\ \hline \text { Finland } & 72 & 53 \\ \text { Finland } & 47 & 51 \\ \text { Finland } & 89 & 72 \\ \text { Lapland } & 55 & 48 \\ \text { Lapland } & 64 & 55 \\ \text { Russia } & 50 & 50 \\ \text { Russia } & 41 & 50 \\ \text { Russia } & 55 & 45 \\ \hline \end{array}\) It is hypothesized that in winter, "loner" males (not present in summer packs) join the pack to increase survival rate. Use a \(5 \%\) level of significance to test the claim that the average percentage of males in a wolf pack is higher in winter.

Step by step solution

01

Identify Hypotheses

First, we'll define the null and alternative hypotheses. The null hypothesis (H0) states that there is no difference in the average percentage of males in winter and summer. The alternative hypothesis (H1) states that the average percentage of males is higher in winter than in summer. Mathematically, this means: \[ H_0: \mu_{\text{winter}} = \mu_{\text{summer}} \] \[ H_1: \mu_{\text{winter}} > \mu_{\text{summer}} \] where \( \mu_{\text{winter}} \) and \( \mu_{\text{summer}} \) are the mean percentages of males in winter and summer, respectively.

02

Calculate Sample Means

Calculate the mean percentage of males in winter and summer. From the given data: \[ \text{Winter: } \frac{72 + 47 + 89 + 55 + 64 + 50 + 41 + 55}{8} = 59.125 \% \] \[ \text{Summer: } \frac{53 + 51 + 72 + 48 + 55 + 50 + 50 + 45}{8} = 53 \% \]

03

Calculate Standard Deviations

Calculate the standard deviation for each set of percentages. Use the formula: \[ s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2} \] Calculate separately for winter and summer percentages to get \( s_{\text{winter}} \) and \( s_{\text{summer}} \).

04

Conduct t-test

Perform a paired t-test because the data consists of matched pairs. The t-statistic is given by: \[ t = \frac{\bar{d} - 0}{s_d / \sqrt{n}} \] where \( \bar{d} \) is the mean of the differences between pairs, and \( s_d \) is the standard deviation of differences. Compute these values and determine the t-statistic.

05

Determine Critical t-value

Given a 5% significance level and 7 degrees of freedom (8 pairs - 1), use the t-distribution table to find the critical value for a one-tailed test. Let it be denoted by \( t_{\alpha} \).

06

Compare t-statistic and Critical t-value

Compare the calculated t-statistic from Step 4 with the critical t-value from Step 5. If the t-statistic is greater than the critical t-value, reject the null hypothesis.

07

Conclusion

Based on the comparison, if the null hypothesis is rejected, conclude that the average percentage of males is significantly higher in winter than in summer. Otherwise, there is not enough evidence to support the claim.

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Key Concepts

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

Paired t-test

Picture a scenario where you want to compare two sets of related data. This is where a paired t-test steps in. It's a statistical method used when each data point in one sample is uniquely paired with a data point in another sample.

The environmental wolf pack study is a great example. Researchers have male percentage data from both winter and summer for the same sets of wolf packs. Each pack's winter data is intrinsically linked to its summer data. This makes a paired t-test the right choice.

The paired t-test checks if the mean difference between these two linked sets is zero. In our wolf pack example, this means asking: "Is there a significant difference in male percentages between winter and summer?"

Here's how it works in practice:

• Calculate mean differences for each pair of values (e.g., subtract summer percentage from winter percentage).
• Compute the average of these differences.
• Find the standard deviation of these differences.
• Use these to calculate the t-statistic.

The t-statistic helps determine whether any observed difference in means is meaningful or just due to random chance.

Null and Alternative Hypotheses

Hypotheses are central to hypothesis testing, acting as statements that you aim to test.

Your null hypothesis (H_0) is like a baseline assumption. It proposes that there's no effect or difference—in this case, that the percentage of males in the packs is the same in winter as it is in summer. Mathematically: H_0: \( \mu_{\text{winter}} = \mu_{\text{summer}} \)

Contrasting this is the alternative hypothesis (H_1), which suggests a change or effect exists. For our wolf packs, the alternative hypothesis claims that male percentages are higher in winter. Expressed mathematically, it's: H_1: \( \mu_{\text{winter}} > \mu_{\text{summer}} \)

During hypothesis testing, you're essentially trying to find evidence against the null hypothesis. A large enough evidence against H_0 in favor of H_1 would indicate that the null hypothesis can be rejected.

Hypotheses guide the entire testing process, setting the foundation for data collection, analysis, and inference.

Significance Level

The significance level is a threshold in hypothesis testing. It's like a line in the sand for deciding if your results are statistically significant.

In the wolf pack study, researchers use a 5% significance level, denoted as \( \alpha \). This means that there's a 5% risk of concluding that there is a significant difference when there really isn't any—a false positive.

How is this used? The significance level aids in determining the critical value from the t-distribution, which is compared to the calculated t-statistic.

• If the t-statistic exceeds this critical value, it suggests the observed data is unlikely under the null hypothesis.
• This leads to rejecting the null hypothesis, considering the result "statistically significant."

Conversely, if the t-statistic does not exceed the critical value, you fail to reject the null hypothesis, implying there's insufficient evidence for the claim.

Setting the significance level is crucial since it affects the sensitivity and rigor of your statistical conclusions, balancing the risks of false findings.