To determine the slope ($b$) of the regression line in a simple linear regression analysis, we can use the formula:

$b = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}$

Where:

• $\text{Cov}(X, Y)$ is the covariance between the predictor ($X$) and the response ($Y$).
• $\text{Var}(X)$ is the variance of the predictor ($X$).

Steps:

1. Calculate the mean of $X$ (sleep hours) and $Y$ (GPA).
2. Compute the covariance $\text{Cov}(X, Y)$: $\text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^{n} \left( X_i - \bar{X} \right) \left( Y_i - \bar{Y} \right)$
3. Compute the variance of $X$: $\text{Var}(X) = \frac{1}{n} \sum_{i=1}^{n} \left( X_i - \bar{X} \right)^2$
4. Use these values to calculate $b$.

Let me compute this step by step. numpy as np

Data

sleep_hours = np.array([6, 7, 5, 8, 6, 7, 5, 7, 8, 6]) gpa = np.array([2.8, 3.0, 2.5, 3.5, 2.9, 3.1, 2.7, 3.2, 3.6, 2.9])

Step 1: Means

mean_sleep = np.mean(sleep_hours) mean_gpa = np.mean(gpa)

Step 2: Covariance

cov_xy = np.mean((sleep_hours - mean_sleep) * (gpa - mean_gpa))

Step 3: Variance of X

var_x = np.mean((sleep_hours - mean_sleep)**2)

Step 4: Slope b

slope_b = cov_xy / var_x

mean_sleep, mean_gpa, cov_xy, var_x, slope_b### Results:

1. Mean of sleep hours ($\bar{X}$): $6.5$
2. Mean of GPA ($\bar{Y}$): $3.02$
3. Covariance ($\text{Cov}(X, Y)$): $0.32$
4. Variance of sleep hours ($\text{Var}(X)$): $1.05$
5. Slope of the regression line ($b$): $0.305$ (approximately).

Interpretation:

The slope ($b = 0.305$) indicates that for each additional hour of sleep, the predicted GPA increases by about $0.305$ points.

Do you want to proceed with calculating the full regression equation, or do you have any other questions?

Here are 5 questions you might consider next:

1. What is the intercept ($a$) of the regression line?
2. How do we interpret the regression equation in this context?
3. What is the coefficient of determination ($R^2$)?
4. Can we use this model to predict GPA for specific sleep hours (e.g., 6 or 9)?
5. What are the assumptions of linear regression, and are they met in this analysis?

Tip: Always visualize your data (e.g., scatter plot) to assess the relationship before relying on regression results.