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Wildlife: Wolves The following is based on information from The Wolf in the Southwest: The Making of an Endangered Species by David E. Brown (University of Arizona Press). Before \(1918,\) the proportion of female wolves in the general population of all southwestern wolves was about \(50 \% .\) However, after 1918, southwestern cattle ranchers began a widespread effort to destroy wolves. In a recent sample of 34 wolves, there were only 10 females. One theory is that male wolves tend to return sooner than females to their old territories where their predecessors were exterminated. Do these data indicate that the population proportion of female wolves is now less than \(50 \%\) in the region? Use \(\alpha=0.01.\)

Step by step solution

01

Establish the Hypotheses

We need to determine if the proportion of female wolves is less than 50%. So, our null hypothesis \(H_0\) is that the proportion of female wolves \(p = 0.5\). The alternative hypothesis \(H_1\) is that \(p < 0.5\).

02

Identify Sample Statistics

The sample size \(n\) is 34, and the number of females \(X\) is 10. Therefore, the sample proportion \(\hat{p}\) of female wolves is calculated as \(\hat{p} = \frac{X}{n} = \frac{10}{34} \approx 0.294\).

03

Calculate the Test Statistic

The test statistic for a proportion is given by \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\), where \(p_0 = 0.5\). Substitute the values to get \(z = \frac{0.294 - 0.5}{\sqrt{\frac{0.5\times 0.5}{34}}}\).

04

Compute the Test Statistic Value

Calculating further, we find \(z \approx \frac{-0.206}{0.086}\approx -2.395\).

05

Determine the Critical Value

For a one-tailed test with \(\alpha = 0.01\), the critical value from the Z-table is approximately -2.33.

06

Make a Decision

Since the calculated \(z\)-value of about -2.395 is less than the critical value of -2.33, we reject the null hypothesis \(H_0\).

07

Conclusion

There is sufficient evidence at the \(\alpha = 0.01\) significance level to conclude that the population proportion of female wolves is less than 50%.

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Key Concepts

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

Proportion Test

The proportion test is a type of hypothesis test used to determine if the proportion of a certain attribute in a population is equal to a specified value. In our wolf example, we are interested in whether the proportion of female wolves is less than 50%. We do this by comparing the proportion derived from our sample data with a hypothesized population proportion.

A proportion test is used when the outcome is binary, meaning each case either has the attribute or does not. It involves:

• Selecting a sample from the population.
• Calculating the sample proportion.
• Conducting the test to compare the sample proportion to the population proportion.

This method helps us understand how our sample relates to the whole population.

Z-Score

The Z-score is a statistical measure that tells us how far away a data point is from the mean of the data set, expressed in terms of standard deviations. In hypothesis testing, the Z-score is used to determine the stance of the sample proportion compared to the hypothesized population proportion, taking into account sampling variability.

For our wolf population, after identifying the sample statistics, we calculate the Z-score to see how far the sample proportion of female wolves is from 0.5, the hypothesized proportion. The formula used is \[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]where:

• \(\hat{p}\) is the sample proportion,
• \(p_0\) is the hypothesized population proportion,
• \(n\) is the sample size.

A higher absolute value of the Z-score suggests a stronger evidence against the null hypothesis.

Significance Level

The significance level, denoted by \(\alpha\), is the threshold for deciding whether to reject the null hypothesis. It defines the probability of making a Type I error, which is rejecting a true null hypothesis. In simpler terms, it's how willing we are to be wrong if we claim there is an effect when, in fact, there isn't.

In our exercise, the significance level is set at 0.01, implying we are 99% confident in the results.
This strict level means that we require strong evidence against the null hypothesis to reject it. Lower significance levels demand more conclusive evidence, reducing the risk of making a Type I error.

Null Hypothesis

The null hypothesis (often symbolized as \(H_0\)) represents the baseline or default statement about a population parameter that there's no effect or no difference. It is what we attempt to test against using statistical methods.

In the case of the wolves, the null hypothesis is that the proportion of female wolves is 50%, or \(H_0: p = 0.5\). We run a hypothesis test to determine if there is sufficient evidence to support the claim that the actual proportion is less than 50%.
The beauty of hypothesis testing is that it provides a structured way to make decisions based on data. By analyzing sample evidence in light of the null hypothesis, we can make more informed conclusions about the larger population.