Problem Breakdown:

We are asked to find the equation of the regression line based on the number of hours spent studying (independent variable $x$) and the corresponding midterm grades (dependent variable $y$). This involves calculating:

1. The slope of the regression line ($b_1$).
2. The y-intercept ($b_0$).
3. The correlation coefficient to check if the relationship between hours spent studying and midterm grades is statistically significant.

Step 1: Gather the data points

• $x$ (Hours Studying): $0, 0.5, 1, 2, 2.5, 3, 3.5, 4, 4.5, 5$
• $y$ (Midterm Grades): $62, 75, 77, 79, 80, 81, 83, 88, 92, 98$

Step 2: Calculate the Slope ($b_1$)

The slope formula is: $b_1 = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$

Step 3: Calculate the Intercept ($b_0$)

The intercept formula is: $b_0 = \bar{y} - b_1 \bar{x}$ where $\bar{x}$ and $\bar{y}$ are the means of the $x$ and $y$ data points.

Step 4: Calculate the Correlation Coefficient $r$

The correlation coefficient $r$ can be calculated using: [ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} ] $r$ will help us determine if the relationship between $x$ and $y$ is statistically significant.

I'll calculate these now.import numpy as np

Data points for hours spent studying (x) and corresponding midterm grades (y)

x = np.array([0, 0.5, 1, 2, 2.5, 3, 3.5, 4, 4.5, 5]) y = np.array([62, 75, 77, 79, 80, 81, 83, 88, 92, 98])

Number of data points

n = len(x)

Calculate sums and means

sum_x = np.sum(x) sum_y = np.sum(y) sum_xy = np.sum(x * y) sum_x_squared = np.sum(x2) sum_y_squared = np.sum(y2)

mean_x = np.mean(x) mean_y = np.mean(y)

Calculate the slope (b1) and intercept (b0)

b1 = (n * sum_xy - sum_x * sum_y) / (n * sum_x_squared - sum_x**2) b0 = mean_y - b1 * mean_x

Calculate the correlation coefficient (r)

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

b1, b0, r### Results:

1. Regression Line Equation: $\hat{y} = 67.42 + 5.42x$ Where $\hat{y}$ is the predicted midterm grade, and $x$ is the number of hours spent studying.

2. Correlation Coefficient ($r$): $r = 0.939$

Since the correlation coefficient is close to 1, this indicates a strong positive relationship between hours spent studying and midterm grades. Thus, the relationship is likely statistically significant, making it appropriate to use the regression line for prediction.

Would you like more details or have any questions?

Related Questions:

1. How can we interpret the slope in the regression equation?
2. What does the y-intercept represent in the context of this problem?
3. How can we test the statistical significance of the correlation coefficient?
4. Can we use this regression model to predict grades for students studying more than 5 hours?
5. What assumptions are made when performing linear regression?
6. How does the correlation coefficient change with different datasets?
7. How do outliers affect the regression line and correlation coefficient?
8. Can we determine the residuals for this regression model?

Tip:

A high correlation coefficient doesn't necessarily imply causation; always consider the context before making predictions.