To calculate the coefficient of determination ($r^2$), we first need to follow these steps:

1. Find the correlation coefficient $r$, which can be calculated using the formula:

$r = \frac{n\sum{xy} - \left(\sum{x}\right)\left(\sum{y}\right)}{\sqrt{ \left(n\sum{x^2} - \left(\sum{x}\right)^2\right) \left(n\sum{y^2} - \left(\sum{y}\right)^2\right) }}$

Where:

• $n$ is the number of data points,
• $x$ is the independent variable (Age),
• $y$ is the dependent variable (Bone Density),
• $\sum{xy}$ is the sum of the product of each pair of $x$ and $y$,
• $\sum{x}$, $\sum{y}$, $\sum{x^2}$, and $\sum{y^2}$ are the sums of the individual $x$ values, $y$ values, squared $x$ values, and squared $y$ values, respectively.

2. Square the correlation coefficient ($r^2$) to get the coefficient of determination.

Let's first calculate all necessary components for the formula.

Given Data:

• Age ($x$): 37, 41, 65, 69, 73
• Bone Density ($y$): 355, 335, 330, 320, 315

We will now compute the necessary sums:

• $\sum{x} = 37 + 41 + 65 + 69 + 73 = 285$
• $\sum{y} = 355 + 335 + 330 + 320 + 315 = 1655$
• $\sum{x^2} = 37^2 + 41^2 + 65^2 + 69^2 + 73^2 = 1369 + 1681 + 4225 + 4761 + 5329 = 23665$
• $\sum{y^2} = 355^2 + 335^2 + 330^2 + 320^2 + 315^2 = 126025 + 112225 + 108900 + 102400 + 99225 = 549775$
• $\sum{xy} = (37)(355) + (41)(335) + (65)(330) + (69)(320) + (73)(315) = 13135 + 13735 + 21450 + 22080 + 22995 = 93395$

Step 1: Calculate $r$

We will now plug the sums into the correlation coefficient formula.

$r = \frac{5 \times 93395 - (285 \times 1655)}{\sqrt{ \left(5 \times 23665 - 285^2\right) \left(5 \times 549775 - 1655^2\right) }}$

After performing these calculations:

$r = \frac{466975 - 471075}{\sqrt{(118325 - 81225)(2748875 - 2747025)}}$

Now simplify:

$r = \frac{-5100}{\sqrt{37100 \times 1850}} = \frac{-5100}{\sqrt{68535000}} \approx \frac{-5100}{8267.53} \approx -0.617$

Step 2: Calculate $r^2$

$r^2 = (-0.617)^2 = 0.381$

Final Answer:

The coefficient of determination $r^2$ is 0.381.

Let me know if you need further clarification or details! Here's a list of related questions:

1. What does the coefficient of determination represent in this context?
2. How do we interpret the value of $r^2$?
3. What is the formula for calculating the correlation coefficient $r$ and why is it important?
4. How can the coefficient of determination help in predicting bone density based on age?
5. What are other statistical measures used to analyze relationships between variables?

Tip: Always check the units and context of your variables when interpreting the coefficient of determination!