To calculate the correlation coefficient $r$, we use the formula:

$r = \frac{n \sum xy - \left( \sum x \right) \left( \sum y \right)}{\sqrt{\left( n \sum x^2 - \left( \sum x \right)^2 \right) \left( n \sum y^2 - \left( \sum y \right)^2 \right)}}$

Where:

• $n$ is the number of data points (in this case, $n = 10$),
• $x$ represents the hours spent studying,
• $y$ represents the midterm grades.

Data:

$x = [0, 0.5, 2, 2.5, 3, 4, 4.5, 5, 5.5, 6]$ $y = [60, 63, 66, 69, 78, 81, 84, 90, 93, 96]$

Steps:

1. Calculate the necessary summations: $\sum x$, $\sum y$, $\sum x^2$, $\sum y^2$, and $\sum xy$.
2. Substitute these values into the formula to compute $r$.

Let me calculate this for you. numpy as np

Given data

x = np.array([0, 0.5, 2, 2.5, 3, 4, 4.5, 5, 5.5, 6]) y = np.array([60, 63, 66, 69, 78, 81, 84, 90, 93, 96])

Calculate the necessary sums

n = len(x) 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)

Calculate the correlation coefficient (r)

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

r_rounded = round(r, 3) r_roundedThe correlation coefficient $r$ is approximately 0.981 (rounded to three decimal places).

This suggests a very strong positive linear relationship between the number of hours spent studying and the midterm grades.

Would you like more details or have any questions about this calculation?

Here are five related questions for further exploration:

1. What does a correlation coefficient of 0.981 indicate about the strength of the relationship?
2. How do you interpret a negative correlation coefficient?
3. How can you use the correlation coefficient to predict future values in this dataset?
4. What is the formula for calculating the coefficient of determination (R-squared), and how is it related to the correlation coefficient?
5. How can outliers affect the correlation coefficient?

Tip: The closer the value of $r$ is to 1 or -1, the stronger the relationship between the variables. A positive value indicates a positive relationship, while a negative value would indicate a negative relationship.