When an anthropologist finds skeletal remains, they need to figure out the height of the person. The height of a person (in cm) and the length of their metacarpal bone 1 (in mm) were collected and are in the table. Find the correlation coefficient and coefficient of determination and then interpret both.

Data of Metacarpal versus Height
Length of
Metacarpal
(mm)	Height of
Person
(cm)
48	178
45	165
41	170
39	162
42	163
47	170
51	178
46	174
44	158


The correlation coefficient is r = (round to five decimal places), which is a Correct , Correct correlation.
The coefficient of determination is r2 =  (round to four decimal places), so % of the variability in the height of a person is explained by the length of the metacarpal. 

To solve this, we need to calculate both the correlation coefficient (\(r\)) and the coefficient of determination (\(r^2\)) for the given data.

### Steps:
1. **Calculate the correlation coefficient** \( r \):
   - This measures the strength and direction of a linear relationship between the two variables: length of metacarpal bone and height of person.
   
2. **Calculate the coefficient of determination** \( r^2 \):
   - This indicates the percentage of the variability in one variable that can be explained by the other variable.

Let's perform the calculation.

Given dataTo solve this, we need to calculate both the correlation coefficient (\(r\)) and the coefficient of determination (\(r^2\)) for the given data.

### Steps:
1. **Calculate the correlation coefficient** \( r \):
   - This measures the strength and direction of a linear relationship between the two variables: length of metacarpal bone and height of person.
   
2. **Calculate the coefficient of determination** \( r^2 \):
   - This indicates the percentage of the variability in one variable that can be explained by the other variable.

Let's perform the calculation.

Given 

| Metacarpal Length (mm) | Height (cm) |
|-----------------------|-------------|
| 48                    | 178         |
| 45                    | 165         |
| 41                    | 170         |
| 39                    | 162         |
| 42                    | 163         |
| 47                    | 170         |
| 51                    | 178         |
| 46                    | 174         |
| 44                    | 158         |

I will now calculate the values.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the correlation coefficient and coefficient of determination using the following steps:

1. **Calculate the correlation coefficient** using the formula:
   \[
   r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{\left[n \sum x^2 - (\sum x)^2\right] \left[n \sum y^2 - (\sum y)^2\right]}}
   \]
   Where:
   - \(x\) is the metacarpal length
   - \(y\) is the height

2. **Coefficient of determination** \( r^2 \) is simply the square of the correlation coefficient:
   \[
   r^2 = (r)^2
   \]

Would you like more guidance on how to manually perform these calculations, or do you have any further questions?

---

Here are some related questions to consider:
1. What does a positive vs. negative correlation coefficient indicate?
2. How do you interpret a coefficient of determination of 0.9?
3. Can correlation coefficients imply causation between two variables?
4. What are the limitations of using correlation for small datasets?
5. How does linear regression relate to correlation and determination?

**Tip:** Always ensure your data is in a linear relationship before calculating the correlation coefficient, as non-linear relationships can give misleading results.

When an anthropologist finds skeletal remains, they need to figure out the height of the person. The height of a person (in cm) and the length of their metacarpal bone 1 (in mm) were collected and are in the table. Find the correlation coefficient and coefficient of determination and then interpret both.

Learn how to calculate the correlation coefficient and coefficient of determination between the length of a metacarpal bone and a person's height. This problem includes key statistical concepts such as linear regression and correlation. Suitable for college-level statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation