A famous researcher observed that chimpanzees hunt and eat meat as part of their regular diet. Sometimes chimpanzees hunt​ alone, while other times they form hunting parties. The following table summarizes research on chimpanzee hunting​ parties, giving the size of the hunting party and the percentage of successful hunts.
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Part 1
Numbers of Chimps
in Hunting Party
Percentage of
Successful Hunts
1
17
2
27
3
25
4
39
5
37
6
55
7
42
8
55
9
62
10
60
12
72
13
72
14
77
15
72
16
79
Find the correlation coefficient. Combining this with the answer to part​ (a), does the percentage of successful hunts tend to increase with the size of the hunting​ party?
requals

To find the correlation coefficient, \( r \), between the size of the chimpanzee hunting party and the percentage of successful hunts, we need to calculate the covariance of the two variables and then normalize it by their standard deviations. The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables.

The formula for the correlation coefficient \( r \) is:

\[
r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}}
\]

where:
- \( x_i \) is each individual number of chimps in the hunting party.
- \( y_i \) is the corresponding percentage of successful hunts.
- \( \bar{x} \) is the mean of the number of chimps in the hunting party.
- \( \bar{y} \) is the mean of the percentage of successful hunts.

Let's compute the required quantities step-by-step.

### Step 1: Calculate the Means
First, we calculate the mean of each variable:

\[
\bar{x} = \frac{\sum x_i}{n}, \quad \bar{y} = \frac{\sum y_i}{n}
\]

where \( n = 16 \) (the number of data points).

### Step 2: Compute the Sum of Products
Next, we compute the sum of the products of the deviations:

\[
\sum (x_i - \bar{x})(y_i - \bar{y})
\]

### Step 3: Compute the Sum of Squared Deviations
Then, compute the sum of the squared deviations for both \( x \) and \( y \):

\[
\sum (x_i - \bar{x})^2, \quad \sum (y_i - \bar{y})^2
\]

### Step 4: Compute \( r \)
Finally, plug these values into the formula for \( r \).

Let's compute this using the data provided.The correlation coefficient \( r \) is approximately **0.963**.

### Conclusion
Since the correlation coefficient \( r = 0.963 \) is very close to 1, there is a strong positive linear relationship between the size of the hunting party and the percentage of successful hunts. This means that as the size of the chimpanzee hunting party increases, the percentage of successful hunts tends to increase as well.

Would you like more details on any of the steps, or do you have any other questions?

Here are 5 additional related questions:
1. How would the correlation coefficient change if outliers were present in the data?
2. What does a correlation coefficient close to 0 indicate about two variables?
3. How can we visually represent the relationship between the size of the hunting party and the percentage of successful hunts?
4. What are some factors that could influence the success rate of chimpanzee hunting parties other than party size?
5. How can we predict the percentage of successful hunts for a given party size using a regression model?

**Tip:** When interpreting correlation, remember that correlation does not imply causation. Even if two variables are correlated, it does not mean that one causes the other.

Learn how to calculate the correlation coefficient between the size of chimpanzee hunting parties and their success rates. Understand the strong positive relationship observed, supported by statistical analysis. Suitable for advanced high school or college students.

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