Find the correlation coefficient,

r

,

of the data described below.

A medical student reads that with enough sun exposure, the human body can manufacture vitamin D3. This sounds too amazing to be true, so she decides to test it out.

Over the next few weeks, she spends a varying number of hours in the sun, x. At the end of each week, she measures the concentration of vitamin D3 in her blood (in nanograms per milliliter), y.

Hours spent in sun

Concentration of vitamin D3 (in ng per ml)

3

18

7

33

8

22

10

58

10

25

*Round* *your answer to the nearest* *thousandth.*

r

=

To find the correlation coefficient \( r \), we need to 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,
- \( x \) is the number of hours spent in the sun,
- \( y \) is the concentration of vitamin D3,
- \( \sum{xy} \) is the sum of the products of corresponding \( x \) and \( y \),
- \( \sum{x} \) and \( \sum{y} \) are the sums of \( x \) and \( y \) values respectively,
- \( \sum{x^2} \) and \( \sum{y^2} \) are the sums of the squares of \( x \) and \( y \) values respectively.

We have the following dataTo find the correlation coefficient \( r \), we need to 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,
- \( x \) is the number of hours spent in the sun,
- \( y \) is the concentration of vitamin D3,
- \( \sum{xy} \) is the sum of the products of corresponding \( x \) and \( y \),
- \( \sum{x} \) and \( \sum{y} \) are the sums of \( x \) and \( y \) values respectively,
- \( \sum{x^2} \) and \( \sum{y^2} \) are the sums of the squares of \( x \) and \( y \) values respectively.

We have the following 

\[
x = [3, 7, 8, 10, 10]
\]
\[
y = [18, 33, 22, 58, 25]
\]

Now, let's compute \( r \).The correlation coefficient \( r \) is approximately \( 0.573 \), rounded to the nearest thousandth.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How does the correlation coefficient \( r \) interpret the strength of the relationship?
2. What would a correlation coefficient of \( r = 1 \) indicate in this context?
3. Can a negative \( r \) value occur in this type of experiment, and what would it signify?
4. How can outliers affect the value of \( r \)?
5. What is the difference between correlation and causation in statistical analysis?

**Tip:** A correlation coefficient close to \( +1 \) or \( -1 \) indicates a stronger relationship between variables, while a value near 0 indicates a weak relationship.

Find the correlation coefficient, r, of the data described below. A medical student reads that with enough sun exposure, the human body can manufacture vitamin D3. This sounds too amazing to be true, so she decides to test it out. Over the next few weeks, she spends a varying number of hours in the sun, x. At the end of each week, she measures the concentration of vitamin D3 in her blood (in nanograms per milliliter), y. Hours spent in sun: 3, 7, 8, 10, 10. Concentration of vitamin D3 (in ng per ml): 18, 33, 22, 58, 25. Round your answer to the nearest thousandth.

Learn how to calculate the correlation coefficient (r) for a medical experiment testing the relationship between hours of sun exposure and vitamin D3 concentration in the blood. This detailed solution applies statistical methods and the correlation coefficient formula, suitable for high school students in Grades 10-12.

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