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.