To solve for covariance and the correlation coefficient, we will use the following formulas:

Covariance Formula:

$\text{Cov}(X, Y) = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{n-1}$ Where $\bar{X}$ and $\bar{Y}$ are the means of $X$ and $Y$, and $n$ is the number of data points.

Correlation Coefficient Formula:

$r = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y}$ Where $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$.

Now, let’s compute the np import pandas as pd

Data

hours_studied = np.array([1, 9, 4, 5, 10, 3, 2, 9, 3, 6, 9]) test_grades = np.array([45, 93, 70, 84, 100, 56, 58, 91, 66, 76, 95])

Mean of X and Y

mean_x = np.mean(hours_studied) mean_y = np.mean(test_grades)

Covariance calculation

covariance = np.sum((hours_studied - mean_x) * (test_grades - mean_y)) / (len(hours_studied) - 1)

Standard deviations of X and Y

std_x = np.std(hours_studied, ddof=1) std_y = np.std(test_grades, ddof=1)

Correlation coefficient calculation

correlation_coefficient = covariance / (std_x * std_y)

Results

covariance, correlation_co### Results: a. Covariance: 56.91
b. Correlation Coefficient: 0.963 (rounded to three decimal places)

The high positive correlation coefficient indicates a strong positive relationship between hours studied and test grades.

Do you want details on the calculations or any specific aspect explained further?
Here are five related questions to expand your understanding:

1. What does a correlation coefficient close to 1 imply in this context?
2. How is covariance different from correlation?
3. Can this relationship be represented as a linear equation?
4. What would a negative correlation coefficient indicate?
5. How does increasing the sample size affect covariance and correlation?

Tip: Always plot the data (scatterplot) to visually inspect the relationship between variables. This can confirm the strength and type of correlation