Calculate and report the absolute value of the "correlation coefficient" for the following Regression table:

 	df	SS	MS	F
Regression	1	33	33	44
Residual	20	15	0.75	
Total	21	48	 	 

**Round to two decimal places**
For example, 45.67% is to be reported as 0.4567

To calculate the absolute value of the correlation coefficient \( |r| \), we first need to find the coefficient of determination \( R^2 \) from the given regression table, which is related to the correlation coefficient as \( r = \sqrt{R^2} \).

Here are the steps:

### Step 1: Calculate \( R^2 \)
The coefficient of determination \( R^2 \) is the proportion of the total variation explained by the regression model. It is calculated as:
\[
R^2 = \frac{\text{SS}_{\text{Regression}}}{\text{SS}_{\text{Total}}}
\]

From the table, we have:
- \( \text{SS}_{\text{Regression}} = 33 \)
- \( \text{SS}_{\text{Total}} = 48 \)

So, \( R^2 \) is:
\[
R^2 = \frac{33}{48} = 0.6875
\]

### Step 2: Calculate the correlation coefficient \( r \)
Since \( r = \sqrt{R^2} \), we can now calculate \( r \):
\[
r = \sqrt{0.6875} \approx 0.8292
\]

### Step 3: Report the absolute value of the correlation coefficient
The absolute value of \( r \) is:
\[
|r| = 0.8292
\]

Thus, the absolute value of the correlation coefficient is **0.8292**.

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. What is the significance of the correlation coefficient in a regression model?
2. How is the \( F \)-statistic related to the significance of the regression model?
3. What does the residual sum of squares (SS) tell us about the model's fit?
4. How do we interpret the coefficient of determination \( R^2 \) in practical terms?
5. What are some assumptions behind linear regression models?

**Tip:** Always check if your regression model satisfies key assumptions like linearity and homoscedasticity before interpreting the results.

Calculate and report the absolute value of the 'correlation coefficient' for the following Regression table:

 df   SS    MS    F
Regression   1   33   33   44
Residual     20   15   0.75  
Total         21   48

**Round to two decimal places**

Learn how to calculate and report the absolute value of the correlation coefficient based on a regression table. This problem covers key statistical concepts such as regression analysis, the coefficient of determination (R²), and the correlation coefficient. The solution is suitable for college-level students.

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