91Ó°ÊÓ

The article "Cooperative Hunting in Lions: The Role of the Individual" (Behavioral Ecology and Sociobiology [1992]: \(445-454\) ) discusses the different roles taken by lionesses as they attack and capture prey. The authors were interested in the effect of the position in line as stalking occurs; an individual lioness may be in the center of the line or on the wing (end of the line) as they advance toward their prey. In addition to position, the role of the lioness was also considered. A lioness could initiate a chase (be the first one to charge the prey), or she could participate and join the chase after it has been initiated. Data from the article are summarized in the accompanying table. $$ \begin{array}{l|cc} & \text { Role } \\ \hline \text { Position } & \text { Initiate Chase } & \text { Participate in Chase } \\ \hline \text { Center } & 28 & 48 \\ \text { Wing } & 66 & 41 \\ \hline \end{array} $$ Is there evidence of an association between position and role? Test the relevant hypotheses using \(\alpha=.01\). What assumptions about how the data were collected must be true for the chi-square test to be an appropriate way to analyze these data?

Step by step solution

01

State the Hypotheses

The null hypothesis is that the two variables (position and role) are independent. The alternative hypothesis is that the variables are not independent, meaning, there is an association between the position and role. Mathematically, these can be stated as: \n\nNull Hypothesis (H0): Position and Role are independent.\n\nAlternative Hypothesis (H1): Position and Role are not independent.

02

Compute the expected frequencies

Each expected frequency can be calculated by the formula (row total * column total) / overall total. \n\nCenter Initiate: \((28 + 48) * (28 + 66) / 183 = 47.14\)\n\nCenter Participate: \((28 + 48) * (48 + 41) / 183 = 28.86\)\n\nWing Initiate: \((66 + 41) * (28 + 66) / 183 = 46.86\)\n\nWing Participate: \((66 + 41) * (48 + 41) / 183 = 60.14\)

03

Compute the chi-square statistic

The chi-square statistic can be computed using the formula: \(\chi^2 = Σ [ (O - E)^2 / E ]\). Here, O denotes observed frequency and E denotes expected frequency. Compute this for all four categories.

04

Determine P-value and conclude

Find the p-value for the chi-square with 1 degree of freedom, then compare the computed chi-square value with table value at 1% level of significance (α=.01), or directly compare p-value with 0.01 to determine whether to accept or reject the null hypothesis

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Statistical Independence

Statistical independence is a fundamental concept in statistics, which refers to the scenario where two variables do not affect or influence each other. In simpler terms, if the occurrence of one event does not change the likelihood of the other, they are considered independent.
In this exercise, we're looking at whether the position of a lioness (Center or Wing) impacts their role in a hunt (Initiate or Participate). If these two factors are independent, it means that knowing a lioness's position doesn't help predict her role in the hunt.

• Null Hypothesis (H0): This states that the position and role are independent. In our lion example, if true, it suggests a lioness's location in the line (Center or Wing) does not influence her decision to Initiate or Participate in the chase.
• Alternative Hypothesis (H1): Contrarily, this hypothesis suggests that there is an association. If the null hypothesis is rejected, it would mean a connection exists between their position and role.

Statistical independence is tested using various tools, the Chi-Square Test being one of the most common ones.

Hypothesis Testing

Hypothesis testing is a method in statistics to determine if there is enough evidence to reject a null hypothesis, based on sample data. This exercise involves conducting a hypothesis test to assess if there's a link between lioness positions and their roles during a hunt.
Hypothesis testing unfolds in a series of steps:

• State the Hypotheses: We start with the null hypothesis (H0), which claims independence, and the alternative hypothesis (H1), which suggests dependence or association.
• Choose Significance Level (α): In this problem, α is set at 0.01, meaning we are looking for evidence strong enough to reject the null hypothesis 99 out of 100 times.
• Compute the Test Statistic: Use the Chi-Square formula to calculate the test statistic, which is then compared to a critical value to decide the outcome of the hypothesis test.
• Make a Decision: Based on the comparison, either reject the null hypothesis or fail to reject it, thus concluding the test.

Hypothesis testing helps ascertain relations between variables and is crucial in making informed conclusions from data.

Expected Frequencies

Expected frequencies are a key component in the Chi-Square Test, indicating what the frequency distribution would look like if the null hypothesis were true. They allow us to measure deviations between the observed and expected data.
To calculate expected frequencies:

• Use the formula: \(E = \frac{\text{(row total)} \times \text{(column total)}}{\text{overall total}}\)
• For example, the expected frequency for 'Center Initiate' is \(\frac{(28 + 48) \times (28 + 66)}{183} = 47.14\).

The expected frequency helps you identify if the observed frequencies are significantly different from what is expected. This difference is used to compute the Chi-Square statistic. If the observed frequencies significantly differ from the expected ones, it suggests that there's an association between the variables, potentially leading to the rejection of the null hypothesis.