To calculate the correlation coefficient $r$, we use the formula:

[ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} ]

Where:

• $n$ is the number of data points (in this case, $n = 5$),
• $x$ represents the values for age,
• $y$ represents the values for bone density,
• $\sum x$ is the sum of the ages,
• $\sum y$ is the sum of the bone densities,
• $\sum xy$ is the sum of the products of age and bone density for each data pair,
• $\sum x^2$ is the sum of the squares of the ages,
• $\sum y^2$ is the sum of the squares of the bone densities.

Given data:

• Age ($x$): 37, 49, 57, 61, 65
• Bone Density ($y$): 360, 355, 330, 320, 310

Let’s break down the necessary calculations.

Step-by-step calculations:

1. Sum of $x$ (ages): $\sum x = 37 + 49 + 57 + 61 + 65 = 269$

2. Sum of $y$ (bone densities): $\sum y = 360 + 355 + 330 + 320 + 310 = 1675$

3. Sum of $x^2$ (squares of ages): $\sum x^2 = 37^2 + 49^2 + 57^2 + 61^2 + 65^2 = 1369 + 2401 + 3249 + 3721 + 4225 = 11965$

4. Sum of $y^2$ (squares of bone densities): $\sum y^2 = 360^2 + 355^2 + 330^2 + 320^2 + 310^2 = 129600 + 126025 + 108900 + 102400 + 96100 = 563025$

5. Sum of $xy$ (product of age and bone density): $\sum xy = (37 \times 360) + (49 \times 355) + (57 \times 330) + (61 \times 320) + (65 \times 310)$ $\sum xy = 13320 + 17395 + 18810 + 19520 + 20150 = 89295$

Now, let's plug these into the formula:

[ r = \frac{5 \times 89295 - 269 \times 1675}{\sqrt{[5 \times 11965 - (269)^2][5 \times 563025 - (1675)^2]}} ]

Simplifying the numerator:

$5 \times 89295 = 446475$ $269 \times 1675 = 450575$ $446475 - 450575 = -4100$

Simplifying the denominator:

1. $5 \times 11965 = 59825$ $(269)^2 = 72361$ $59825 - 72361 = -12536$

2. $5 \times 563025 = 2815125$ $(1675)^2 = 2805625$ $2815125 - 2805625 = 9500$

Now, the denominator is:

$\sqrt{(-12536) \times 9500} = \sqrt{119592000} \approx 10935.15$

Finally, calculating $r$:

$r = \frac{-4100}{10935.15} \approx -0.375$

Final answer:

The correlation coefficient $r$ is approximately -0.375.

Would you like further details or clarifications? Here are some related questions you might find interesting:

1. How do the values of $r$ indicate the strength and direction of a relationship between two variables?
2. What are some other methods to measure the relationship between variables?
3. How does the calculation of correlation change with more data points?
4. Can the correlation coefficient be used for nonlinear relationships?
5. What are the limitations of the correlation coefficient in statistical analysis?

Tip: The correlation coefficient ranges from -1 to 1. A value closer to 1 indicates a strong positive relationship, closer to -1 indicates a strong negative relationship, and around 0 means no linear relationship.