The table below gives the number of hours ten randomly selected students spent studying and their corresponding midterm exam grades. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the midterm exam grade that a student will earn based on the number of hours spent studying. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.

Hours Studying	0	0.5	1	2	2.5	3	3.5	4	4.5	5
Midterm Grades	62	75	77	79	80	81	83	88	92	98

### 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(x**2)
sum_y_squared = np.sum(y**2)

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_x**2) * (n * sum_y_squared - sum_y**2))
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.

Learn how to calculate the regression line equation and correlation coefficient using a set of midterm grades and hours spent studying data. Understand the statistical significance and implications of the correlation coefficient in predicting midterm grades based on study hours.

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