Research was conducted on the amount of oxygen runners could utilize during training, known as their 
 max, and their finishing times for a 5K race to see if there was a significant relationship between these variables. The results for 8 runners were compiled and placed in the following table.

 Max (ml/kg/min) (x)	5K Finishing Time (min) (y)
43.68	23.26
39.4	23.39
38.02	28.98
39.94	26.57
23.67	31.36
41.97	25.4
39.08	27.35
39.55	23.74
Calculate the correlation coefficient. Round to three decimals.

r = 
 
Find the equation of the regression line. Round each value to two decimals.

### Step 1: Organize the Data

Given data for 8 runners:
\[
\begin{array}{|c|c|}
\hline
\text{Max }(x) & \text{Finishing Time }(y) \\
\hline
43.68 & 23.26 \\
39.4  & 23.39 \\
38.02 & 28.98 \\
39.94 & 26.57 \\
23.67 & 31.36 \\
41.97 & 25.4  \\
39.08 & 27.35 \\
39.55 & 23.74 \\
\hline
\end{array}
\]

### Step 2: Formulas

1. **Correlation Coefficient (r)**:
   \[
   r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}
   \]
   where \( n \) is the number of data points (8 in this case).

2. **Regression Line**:
   The equation of the regression line is \( y = mx + b \), where:
   \[
   m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}
   \]
   \[
   b = \frac{\Sigma y - m(\Sigma x)}{n}
   \]

Now, I'll calculate \( r \) and the equation of the regression line.### Results:

1. **Correlation Coefficient (r)**: \( r = -0.800 \) (rounded to three decimals)
   - This negative correlation suggests that as the maximum oxygen utilization (max \( \text{VO}_2 \)) increases, the finishing time tends to decrease, which aligns with the expectation that better cardiovascular fitness is associated with faster running times.

2. **Equation of the Regression Line**:
   \[
   y = -0.38x + 40.74
   \]
   (rounded to two decimals)

Would you like more details, or have any questions?

### Related Questions:
1. How can the correlation coefficient help in interpreting the strength of the relationship?
2. What is the significance of a negative slope in the regression equation?
3. How would you predict the 5K time for a runner with a max \( \text{VO}_2 \) of 45 ml/kg/min?
4. What would the finishing time be if the max \( \text{VO}_2 \) is 30 ml/kg/min using this model?
5. How can you assess the reliability of this regression line for predicting 5K times for other runners?

### Tip:
When analyzing the correlation coefficient, values closer to -1 or 1 indicate stronger relationships, while values near 0 suggest weaker relationships.

Research was conducted on the amount of oxygen runners could utilize during training, known as their max, and their finishing times for a 5K race to see if there was a significant relationship between these variables. The results for 8 runners were compiled and placed in the following table.

Max (ml/kg/min) (x)	5K Finishing Time (min) (y)
43.68	23.26
39.4	23.39
38.02	28.98
39.94	26.57
23.67	31.36
41.97	25.4
39.08	27.35
39.55	23.74

Calculate the correlation coefficient. Round to three decimals.

r = 

Find the equation of the regression line. Round each value to two decimals.

Learn how to calculate the correlation coefficient and the equation of the regression line for a 5K race dataset with 8 runners. The problem focuses on the relationship between max oxygen utilization and 5K finishing times using key statistical formulas. Suitable for college-level students.

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