To calculate the correlation coefficient $r$, we can use the following 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 (5 in this case),
• $x$ represents the age values,
• $y$ represents the bone density values,
• $\sum$ represents the summation over all data points.

Step 1: Organize the Data

The data provided is as follows:

Age $x$	Bone Density $y$
37	355
41	350
45	335
65	330
69	320

Step 2: Calculate the necessary sums

First, calculate the following:

1. $\sum{x}$
2. $\sum{y}$
3. $\sum{xy}$
4. $\sum{x^2}$
5. $\sum{y^2}$

Let’s compute these values:

• $\sum{x} = 37 + 41 + 45 + 65 + 69 = 257$
• $\sum{y} = 355 + 350 + 335 + 330 + 320 = 1690$
• $\sum{xy} = (37 \times 355) + (41 \times 350) + (45 \times 335) + (65 \times 330) + (69 \times 320)$
• $\sum{x^2} = (37^2) + (41^2) + (45^2) + (65^2) + (69^2)$
• $\sum{y^2} = (355^2) + (350^2) + (335^2) + (330^2) + (320^2)$

Let me compute the individual calculations for these values. Data ages = [37, 41, 45, 65, 69] bone_densities = [355, 350, 335, 330, 320]

Calculate necessary sums

sum_x = sum(ages) sum_y = sum(bone_densities) sum_x2 = sum(x2 for x in ages) sum_y2 = sum(y2 for y in bone_densities) sum_xy = sum(ages[i] * bone_densities[i] for i in range(len(ages)))

Number of data points

n = len(ages)

(sum_x, sum_y, sum_xy, sum_x2, sum_y2)The computed values are:

• $\sum{x} = 257$
• $\sum{y} = 1690$
• $\sum{xy} = 86090$
• $\sum{x^2} = 14061$
• $\sum{y^2} = 572050$

Step 3: Plug the values into the correlation coefficient formula

Now we can substitute these values into the formula for the correlation coefficient $r$:

[ r = \frac{5(86090) - (257)(1690)}{\sqrt{[5(14061) - (257)^2][5(572050) - (1690)^2]}} ]

Let me compute this for you. math

Calculate the numerator and denominator of the correlation coefficient formula

numerator = (n * sum_xy) - (sum_x * sum_y) denominator = math.sqrt((n * sum_x2 - sum_x2) * (n * sum_y2 - sum_y2))

Calculate the correlation coefficient

r = numerator / denominator round(r, 3)The correlation coefficient $r$ is approximately -0.923.

This indicates a strong negative linear relationship between age and bone density in this sample.

Do you need further explanation or assistance with any other steps? Here are some related questions you might find useful:

1. What does a negative correlation coefficient signify in the context of this study?
2. How do you interpret the strength of the correlation based on the value of $r$?
3. How would you calculate the regression line using this data?
4. How does the correlation coefficient change if we remove the outlier (65 years, 330 bone density)?
5. What is the significance of correlation in medical studies?

Tip: Always visualize the data (scatter plot) when checking the strength of correlation to better understand the relationship.