Question
year: 2011, 2012, 2013, 2014, 2015, 2016, 2017
number of species added: 19, 51, 89, 66, 31, 74, 11
The table shows the number of species added to the United States endangered species list each year for several years. A linear function can be used to model the data.
what is the value of the correlation coefficent (r-value)?
number of species added: 19, 51, 89, 66, 31, 74, 11
The table shows the number of species added to the United States endangered species list each year for several years. A linear function can be used to model the data.
what is the value of the correlation coefficent (r-value)?
Answers
GPT-4o mini
To calculate the correlation coefficient (r-value) for the given data, we can follow these steps:
1. **Listing the data**: You provided the years and the number of species added:
```
Year: 2011, 2012, 2013, 2014, 2015, 2016, 2017
Species added: 19, 51, 89, 66, 31, 74, 11
```
Let's denote years as \( x \) and number of species added as \( y \):
\[
\begin{align*}
x: & \quad 1, 2, 3, 4, 5, 6, 7 \quad \text{(relative years since 2011)} \\
y: & \quad 19, 51, 89, 66, 31, 74, 11
\end{align*}
\]
(Here, I adjusted the \( x \) values to be relative integers representing the years since 2011.)
2. **Calculating the necessary sums**:
- \( n \): number of data points
- \( \sum x \): sum of x values
- \( \sum y \): sum of y values
- \( \sum xy \): sum of the product of \( x \) and \( y \)
- \( \sum x^2 \): sum of the squares of \( x \)
- \( \sum y^2 \): sum of the squares of \( y \)
Let's calculate these values:
\[
\begin{align*}
n & = 7 \\
\sum x & = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 \\
\sum y & = 19 + 51 + 89 + 66 + 31 + 74 + 11 = 341 \\
\sum xy & = (1*19) + (2*51) + (3*89) + (4*66) + (5*31) + (6*74) + (7*11) \\
& = 19 + 102 + 267 + 264 + 155 + 444 + 77 = 1328 \\
\sum x^2 & = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 \\
& = 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140 \\
\sum y^2 & = 19^2 + 51^2 + 89^2 + 66^2 + 31^2 + 74^2 + 11^2 \\
& = 361 + 2601 + 7921 + 4356 + 961 + 5476 + 121 = 18401 \\
\end{align*}
\]
3. **Calculating the correlation coefficient (r)**:
The formula for the correlation coefficient \( r \) is:
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
Substituting the calculated values into the formula:
\[
r = \frac{7(1328) - (28)(341)}{\sqrt{[7(140) - (28)^2][7(18401) - (341)^2]}}
\]
- Calculating the numerator:
\[
7(1328) = 9296 \quad \text{and} \quad 28 \times 341 = 9558 \implies 9296 - 9558 = -262
\]
- Calculating the denominator:
\[
7(140) = 980 \quad \text{and} \quad (28)^2 = 784 \implies 980 - 784 = 196
\]
\[
7(18401) = 128807 \quad \text{and} \quad (341)^2 = 116281 \implies 128807 - 116281 = 11526
\]
Now substituting back into the denominator:
\[
\sqrt{(196)(11526)} = \sqrt{2263996} \approx 1504.67
\]
Finally, putting it all together:
\[
r = \frac{-262}{1504.67} \approx -0.174
\]
Thus, the correlation coefficient \( r \), which indicates the strength and direction of a linear relationship between the years and the species added, is approximately \(-0.174\). This indicates a weak negative correlation.
1. **Listing the data**: You provided the years and the number of species added:
```
Year: 2011, 2012, 2013, 2014, 2015, 2016, 2017
Species added: 19, 51, 89, 66, 31, 74, 11
```
Let's denote years as \( x \) and number of species added as \( y \):
\[
\begin{align*}
x: & \quad 1, 2, 3, 4, 5, 6, 7 \quad \text{(relative years since 2011)} \\
y: & \quad 19, 51, 89, 66, 31, 74, 11
\end{align*}
\]
(Here, I adjusted the \( x \) values to be relative integers representing the years since 2011.)
2. **Calculating the necessary sums**:
- \( n \): number of data points
- \( \sum x \): sum of x values
- \( \sum y \): sum of y values
- \( \sum xy \): sum of the product of \( x \) and \( y \)
- \( \sum x^2 \): sum of the squares of \( x \)
- \( \sum y^2 \): sum of the squares of \( y \)
Let's calculate these values:
\[
\begin{align*}
n & = 7 \\
\sum x & = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 \\
\sum y & = 19 + 51 + 89 + 66 + 31 + 74 + 11 = 341 \\
\sum xy & = (1*19) + (2*51) + (3*89) + (4*66) + (5*31) + (6*74) + (7*11) \\
& = 19 + 102 + 267 + 264 + 155 + 444 + 77 = 1328 \\
\sum x^2 & = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 \\
& = 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140 \\
\sum y^2 & = 19^2 + 51^2 + 89^2 + 66^2 + 31^2 + 74^2 + 11^2 \\
& = 361 + 2601 + 7921 + 4356 + 961 + 5476 + 121 = 18401 \\
\end{align*}
\]
3. **Calculating the correlation coefficient (r)**:
The formula for the correlation coefficient \( r \) is:
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
Substituting the calculated values into the formula:
\[
r = \frac{7(1328) - (28)(341)}{\sqrt{[7(140) - (28)^2][7(18401) - (341)^2]}}
\]
- Calculating the numerator:
\[
7(1328) = 9296 \quad \text{and} \quad 28 \times 341 = 9558 \implies 9296 - 9558 = -262
\]
- Calculating the denominator:
\[
7(140) = 980 \quad \text{and} \quad (28)^2 = 784 \implies 980 - 784 = 196
\]
\[
7(18401) = 128807 \quad \text{and} \quad (341)^2 = 116281 \implies 128807 - 116281 = 11526
\]
Now substituting back into the denominator:
\[
\sqrt{(196)(11526)} = \sqrt{2263996} \approx 1504.67
\]
Finally, putting it all together:
\[
r = \frac{-262}{1504.67} \approx -0.174
\]
Thus, the correlation coefficient \( r \), which indicates the strength and direction of a linear relationship between the years and the species added, is approximately \(-0.174\). This indicates a weak negative correlation.