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When we perform hypothesis testing to analyze data, our first step is to define two opposing statements, called the null and alternative hypotheses. These form the foundation of our investigation:

• The null hypothesis (denoted as \(H_0\)) represents a statement of no effect or no difference. In our exercise involving Southwestern wolves, \(H_0\) states that the proportion of female wolves is still 50%, or \(p = 0.5\).
• The alternative hypothesis (denoted as \(H_a\)) suggests there is an effect or difference. Here, \(H_a\) posits that the proportion of female wolves has decreased, implying \(p < 0.5\).

These hypotheses set the stage for statistical testing. If our data provides enough evidence against the null hypothesis, we may consider accepting the alternative hypothesis. Thus, formulating these hypotheses is crucial as they drive the direction of our analysis.

Once hypotheses are set, the next step is calculating a test statistic that helps us make a decision regarding these hypotheses. For proportions, the test statistic \(z\) is derived from the standard formula:
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]
Here's what each term represents:

• \(\hat{p}\) is the sample proportion; in this example, 10 female wolves out of 34 total wolves gives \(\hat{p} = \frac{10}{34} \approx 0.294\).
• \(p_0\) is the hypothesized population proportion, which is 50% or 0.5 in our case.
• \(n\) is the sample size, which is 34 for the wolves.

By inserting these values into the formula, \(z\) is calculated. In our wolf example, we find \(z \approx -2.40\). This statistic tells us how many standard deviations the sample proportion is from the hypothesized proportion under a normal distribution. Understanding the calculation of \(z\) helps us quantify the difference between our observed sample and the expectations under the null hypothesis.

The significance level, denoted as \(\alpha\), is a threshold set by us to determine how confident we need to be in rejecting the null hypothesis. It helps quantify the probability of making a Type I error, which occurs when we wrongly reject the true null hypothesis.
In this exercise, \(\alpha = 0.01\), which means we accept a very low risk (only 1%) of incorrectly rejecting \(H_0\). This level of significance informs us that strong evidence is required to support the claim that the proportion of female wolves is less than 50%.
By referring to the standard normal (z) distribution table, we find the critical value corresponding to \(\alpha = 0.01\) for a one-tailed test: \(-2.33\). Since our calculated \(z\) statistic of \(-2.40\) is lower than \(-2.33\), it leads us to reject the null hypothesis.
Choosing and understanding a significance level clarifies the threshold for decision-making in hypothesis testing, ensuring we draw reliable conclusions from sample data.