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To solve this problem, we need to calculate the correlation coefficient \( r \) and the coefficient of determination \( R^2 \) based on the bivariate data set. Here's how we can approach it:

### The data set provided:

| x   | y      |
|-----|--------|
| 24  | 106.8  |
| 76.2| -39.5  |
| 46.8| 75.8   |
| 38.9| 57.6   |
| 54.3| 40.4   |

#### Step 1: Calculating the correlation coefficient \( r \)
The correlation coefficient \( r \) is calculated using the formula:

\[
r = \frac{n\sum{xy} - \sum{x}\sum{y}}{\sqrt{(n\sum{x^2} - (\sum{x})^2)(n\sum{y^2} - (\sum{y})^2)}}
\]

where:
- \( n \) is the number of data points.
- \( \sum{xy} \) is the sum of the products of each pair \( x_i y_i \).
- \( \sum{x} \) and \( \sum{y} \) are the sums of \( x \)-values and \( y \)-values respectively.
- \( \sum{x^2} \) and \( \sum{y^2} \) are the sums of the squares of \( x \)-values and \( y \)-values.

#### Step 2: Coefficient of determination \( R^2 \)
The coefficient of determination \( R^2 \) is simply the square of the correlation coefficient \( r \):

\[
R^2 = r^2
\]

Now, let me calculate the values for you!The correlation coefficient \( r \) is approximately \(-0.954\), and the coefficient of determination \( R^2 \) is about \(91.0\%\). This means that about 91% of the variation in \( y \) can be explained by the variation in \( x \).

Would you like further details on the calculations or have any questions?

Here are some related questions you might find interesting:
1. What does a negative correlation coefficient indicate?
2. How do you interpret the value of \( R^2 \)?
3. What are some factors that can affect the strength of a correlation?
4. How do outliers impact the correlation coefficient?
5. What is the difference between correlation and causation?

**Tip:** Always check if there are any outliers in your data, as they can significantly affect the correlation coefficient.

Find the correlation coefficient and coefficient of determination (R²) from the given bivariate data set, and report them to three decimal places.

Learn how to calculate the correlation coefficient and the coefficient of determination (R²) from a given bivariate dataset. This problem involves using key statistical concepts such as Pearson's correlation coefficient and is suitable for high school or introductory college-level statistics students.

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