The problem you are working on requires calculating the coefficient of determination, $r^2$, based on the given data of hours spent studying and corresponding midterm grades. Here's the data:

To calculate the coefficient of determination, we first need to calculate the correlation coefficient $r$. The formula for $r$ involves computing:

1. The mean of the hours spent studying, $\overline{x}$, and the mean of the midterm grades, $\overline{y}$.
2. The numerator (covariance) for $r$.
3. The denominator (product of standard deviations).

Once we get the correlation coefficient $r$, we square it to find the coefficient of determination $r^2$.

I will go ahead and compute that for you now.

numpy as np

Data from the table

hours_spent = np.array([0, 1, 2, 4, 5]) midterm_grades = np.array([66, 75, 84, 87, 90])

Calculate the means of both variables

mean_x = np.mean(hours_spent) mean_y = np.mean(midterm_grades)

Calculate the covariance and standard deviations

cov_xy = np.sum((hours_spent - mean_x) * (midterm_grades - mean_y)) std_x = np.std(hours_spent) std_y = np.std(midterm_grades)

Calculate the correlation coefficient (r)

r = cov_xy / (len(hours_spent) * std_x * std_y)

Coefficient of determination (r^2)

r_squared = r**2

round(r_squared, 3)The coefficient of determination $r^2$ for the given data is approximately 0.876. This means that about 87.6% of the variation in midterm grades can be explained by the number of hours spent studying.

Would you like any further explanations or steps? Here are some additional related questions:

1. How do we interpret the value of $r^2$ in this context?
2. What is the relationship between $r$ and $r^2$?
3. How would you calculate the correlation coefficient $r$ directly?
4. What impact would adding more data points have on $r^2$?
5. How do we perform hypothesis testing on $r^2$?

Tip: Always ensure that the data you use for such calculations is appropriately scaled and doesn't include outliers, as they can significantly affect the results.