The following table gives the number of parking tickets obtained in a semester and the GPAs of 55 randomly selected drivers.

Number of Tickets

1

3

4

5

8

GPA

5

4.5

4

1.5

1

Copy Data

Calculate the coefficient of determination, r2. Round your answer to three decimal places.




To calculate the coefficient of determination \( r^2 \), we need to first determine the **correlation coefficient** \( r \). This is typically done by following these steps:

### Step 1: Organize the data
The given data is:
- Number of tickets: \( [1, 3, 4, 5, 8] \)
- GPA: \( [5, 4.5, 4, 1.5, 1] \)

### Step 2: Calculate the means of \( X \) (Number of Tickets) and \( Y \) (GPA)
The means are calculated using the formula:

\[
\text{Mean of } X = \frac{\sum X}{n}, \quad \text{Mean of } Y = \frac{\sum Y}{n}
\]
Where \( n = 5 \) (since there are 5 data points).

### Step 3: Calculate the necessary components for \( r \):
We will need to compute the following:

1. The sum of the product of each pair \( X_i \times Y_i \).
2. The sum of the squares of each \( X_i \) and \( Y_i \).

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

\[
r = \frac{n\sum (X_i Y_i) - \sum X_i \sum Y_i}{\sqrt{(n\sum X_i^2 - (\sum X_i)^2)(n\sum Y_i^2 - (\sum Y_i)^2)}}
\]

### Step 4: Calculate \( r^2 \)
Once we compute \( r \), we square it to get the **coefficient of determination** \( r^2 \).

Let me do these calculations now.The coefficient of determination \( r^2 \) is **0.830**, rounded to three decimal places.

Would you like more details or have any further questions on this? Here are some related questions that could expand your understanding:

1. How do you interpret the value of \( r^2 \) in terms of the strength of the relationship between two variables?
2. What is the significance of a high or low correlation coefficient \( r \) in data analysis?
3. How does the coefficient of determination relate to the concept of linear regression?
4. What would happen if the data points were perfectly correlated, and what would \( r^2 \) equal in that case?
5. How can the coefficient of determination be used to make predictions about the data?

**Tip:** The closer \( r^2 \) is to 1, the stronger the relationship between the two variables. Conversely, closer to 0 indicates a weak or no linear relationship.

The following table gives the number of parking tickets obtained in a semester and the GPAs of 55 randomly selected drivers. Number of Tickets: 1, 3, 4, 5, 8; GPA: 5, 4.5, 4, 1.5, 1. Calculate the coefficient of determination, r². Round your answer to three decimal places.

Learn how to calculate the coefficient of determination (r²) using the data on parking tickets and GPA. This problem involves computing the correlation coefficient and using it to determine r², which helps assess the relationship between two variables. Suitable for high school students.

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