To find the equation of the regression line, $\hat{y} = b_0 + b_1x$, follow these steps:

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Steps:

1. Identify the regression line formula:

   • $\hat{y} = b_0 + b_1x$, where: $b_1 = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}$ $b_0 = \frac{\sum{y} - b_1\sum{x}}{n}$

2. Extract data:

   • $x$: Hours unsupervised
   • $y$: Overall grades

3. Compute necessary sums:

   • $\sum{x}, \sum{y}, \sum{x^2}, \sum{y^2}, \sum{xy}$

4. Calculate $b_1$ and $b_0$.

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Data:

[ x = [0, 0.5, 1.5, 2.5, 4, 4.5, 6], \quad y = [88, 79, 76, 72, 65, 63, 62] ]

Let's compute the required values.x = np.array([0, 0.5, 1.5, 2.5, 4, 4.5, 6]) # Hours unsupervised y = np.array([88, 79, 76, 72, 65, 63, 62]) # Overall grades

Number of data points

n = len(x)

Calculate 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)

Calculate slope (b1)

b1 = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2)

Calculate intercept (b0)

b0 = (sum_y - b1 * sum_x) / n

b0, b-importing required library after code state reset import numpy as np

Data

x = np.array([0, 0.5, 1.5, 2.5, 4, 4.5, 6]) # Hours unsupervised y = np.array([88, 79, 76, 72, 65, 63, 62]) # Overall grades

Number of data points

n = len(x)

Calculate 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)

Calculate slope (b1)

b1 = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2)

Calculate intercept (b0)

b0 = (sum_y - b1 * sum_x) / n

b0, b### Equation of the Regression Line: The regression line for predicting overall grade average based on the number of hours spent unsupervised is:

$\hat{y} = 83.369 - 4.136x$

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Interpretation:

1. Slope ($b_1 = -4.136$):
   • For every additional hour spent unsupervised, the predicted grade decreases by approximately 4.136 points.
2. Intercept ($b_0 = 83.369$):
   • If a student spends 0 hours unsupervised, the predicted grade is approximately 83.369.

Would you like to analyze the significance of the correlation coefficient or make predictions using this regression line?

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Follow-up Questions:

1. What is the predicted grade for a student spending 3 hours unsupervised?
2. Is the correlation coefficient for this data statistically significant?
3. Can this model be used reliably for predictions?
4. How does the steep negative slope reflect on the data's trend?
5. Are there any outliers that could significantly influence the regression line?

Tip:

Always verify the statistical significance of the correlation coefficient before interpreting or using the regression line for predictions.