Questions: Find the equation for the least squares regression line of the data described below. Coach Clarke is responsible for recruiting male athletes to join the European Masters track and field team. To improve his recruitment strategies, he wants to investigate the connection between an athlete's height and 3000-meter run time. Coach Clarke has recorded the heights of the men on the track and field team (in centimeters), x, and their best 3000-meter times (in minutes), y. Height 3000-meter time 157 8.94 164 8.54 164 8.32 167 7.42 169 7.93 176 7.97 177 7.57 Round your answers to the nearest thousandth.

Solution Steps

First, we calculate the means of the heights \( \bar{x} \) and the 3000-meter times \( \bar{y} \):

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 167.714 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 8.099 \]

Next, we compute the correlation coefficient \( r \):

\[ r = -0.754 \]

We find the numerator for the slope \( \beta \):

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 9490.54 - 7 \cdot 167.714 \cdot 8.099 = -17.183 \]

Then, we calculate the denominator for the slope \( \beta \):

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 197196 - 7 \cdot 167.714^2 = 299.429 \]

Using the numerator and denominator, we find the slope \( \beta \):

\[ \beta = \frac{-17.183}{299.429} = -0.057 \]

Now, we calculate the intercept \( \alpha \):

\[ \alpha = \bar{y} - \beta \bar{x} = 8.099 - (-0.057) \cdot 167.714 = 17.723 \]

Finally, we can express the equation of the least squares regression line:

\[ y = 17.723 - 0.057x \]

Final Answer