The coefficient of determination, $r^2$, is calculated from the correlation coefficient $r$ as:

$r^2 = r^2$

Steps to Calculate $r^2$:

1. Compute the correlation coefficient $r$:

   • Use the formula: $r = \frac{n\sum{xy} - \sum{x}\sum{y}}{\sqrt{\left(n\sum{x^2} - (\sum{x})^2\right) \left(n\sum{y^2} - (\sum{y})^2\right)}}$ Here:
     • $x$ = Hours spent studying
     • $y$ = Midterm grades
     • $n$ = Number of data points

2. Square $r$ to get $r^2$.

I will calculate it step-by-step.

Data:

[ x = [1, 1.5, 2, 2.5, 3.5, 4, 5], \quad y = [60, 66, 75, 84, 90, 93, 96] ]

Calculating necessary values:

• $\sum x, \sum y, \sum x^2, \sum y^2, \sum xy$.

Let me compute these.import numpy as np

Data

x = np.array([1, 1.5, 2, 2.5, 3.5, 4, 5]) y = np.array([60, 66, 75, 84, 90, 93, 96])

Number of data points

n = len(x)

Calculating sums

sum_x = np.sum(x) sum_y = np.sum(y) sum_x2 = np.sum(x2) sum_y2 = np.sum(y2) sum_xy = np.sum(x * y)

Correlation coefficient r

r_numerator = (n * sum_xy) - (sum_x * sum_y) r_denominator = np.sqrt((n * sum_x2 - sum_x2) * (n * sum_y2 - sum_y2)) r = r_numerator / r_denominator

Coefficient of determination r^2

r_squared = r**2

sum_x, sum_y, sum_x2, sum_y2, sum_xy, r, r_squared(19.5, 564, 66.75, 46602, 1686.0, 0.956691463834754, 0.9152585569742845)(19.5, 564, 66.75, 46602, 1686.0, 0.956691463834754, 0.9152585569742845)success### Calculations:

• $\sum x = 19.5$
• $\sum y = 564$
• $\sum x^2 = 66.75$
• $\sum y^2 = 46602$
• $\sum xy = 1686$

The correlation coefficient $r$ is:

$r = 0.957$

The coefficient of determination $r^2$ is:

$r^2 = 0.915$

Final Answer:

The coefficient of determination is $r^2 = 0.915$ (rounded to three decimal places).

Would you like a detailed explanation of the steps or any clarifications?

Follow-up Questions:

1. What is the significance of $r^2$ in this context?
2. How does $r^2$ indicate the strength of the relationship?
3. Can we predict grades based on hours spent studying using this model?
4. What are the potential limitations of this analysis?
5. How would the result change with additional data?

Tip:

Always check for outliers in the data as they can significantly affect the correlation and determination coefficients.