Math Problem Statement

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.

Solution

Step 1: Organize the Data

Given data for 8 runners:

\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.

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Math Problem Analysis

Mathematical Concepts

Statistics
Linear Regression
Correlation Coefficient

Formulas

Correlation coefficient formula: r = [n(Σxy) - (Σx)(Σy)] / sqrt{[nΣx^2 - (Σx)^2][nΣy^2 - (Σy)^2]}
Regression line formula: y = mx + b
Slope formula: m = [n(Σxy) - (Σx)(Σy)] / [nΣx^2 - (Σx)^2]
Intercept formula: b = [Σy - m(Σx)] / n

Theorems

Least Squares Method

Suitable Grade Level

College/University

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