Question

Find the correlation coefficient of the line of best fit for the points (-3, -40), (1, 12), (5, 72), and (7, 137).. these numbers are the points, you're solving for R=0.982 use the coefficient to describe the correlation

Answer 1

To find the correlation coefficient \( r \) for the dataset given by the points \((-3, -40)\), \( (1, 12)\), \( (5, 72)\), and \( (7, 137)\), we can use the formula for the Pearson correlation coefficient, which is:

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

Where:
- \( n \) is the number of data points
- \( x \) and \( y \) are the variables being correlated

### Step 1: Calculate the required sums

1. **Calculate \( n \)**:
\[
n = 4 \quad (\text{since there are 4 points})
\]

2. **Calculate \( \sum x \)**, \( \sum y \)**, \( \sum xy \)**, \( \sum x^2 \)**, and \( \sum y^2 \)**:
- **Points**:
- \( (-3, -40) \)
- \( (1, 12) \)
- \( (5, 72) \)
- \( (7, 137) \)

- **Calculations**:
\[
\sum x = -3 + 1 + 5 + 7 = 10
\]
\[
\sum y = -40 + 12 + 72 + 137 = 181
\]
\[
\sum xy = (-3)(-40) + (1)(12) + (5)(72) + (7)(137) = 120 + 12 + 360 + 959 = 1451
\]
\[
\sum x^2 = (-3)^2 + (1)^2 + (5)^2 + (7)^2 = 9 + 1 + 25 + 49 = 84
\]
\[
\sum y^2 = (-40)^2 + (12)^2 + (72)^2 + (137)^2 = 1600 + 144 + 5184 + 18769 = 25397
\]

### Step 2: Substitute into the formula

Now we can substitute these values into the formula for \( r \):

\[
r = \frac{4(1451) - (10)(181)}{\sqrt{[4(84) - (10)^2][4(25397) - (181)^2]}}
\]

Calculating the components step-by-step:
- Calculating \( 4(1451) - (10)(181) \):
\[
= 5804 - 1810 = 3994
\]

- Calculating \( n \sum x^2 - (\sum x)^2 \):
\[
= 4(84) - (10)^2 = 336 - 100 = 236
\]

- Calculating \( n \sum y^2 - (\sum y)^2 \):
\[
= 4(25397) - (181)^2 = 101588 - 32761 = 68827
\]

Now, substituting these into the formula for \( r \):

\[
r = \frac{3994}{\sqrt{236 \cdot 68827}}
\]

Calculating the denominator:

\[
\sqrt{236 \cdot 68827} \approx \sqrt{16263932} \approx 4032.08
\]

So,

\[
r \approx \frac{3994}{4032.08} \approx 0.989
\]

### Step 3: Correlation Coefficient Description

If we assume you found \( r = 0.982 \), we can describe the correlation as follows:

- The correlation coefficient \( r = 0.982 \) indicates a **very strong positive linear relationship** between the two variables represented by the points.
- This means that as the x-values increase, the y-values tend to also increase, and the data points are closely clustered around the line of best fit.

In summary, a correlation coefficient of \( 0.982 \) suggests a strong predictable linear association between the variables, confirming that approximately 96.4% (i.e., \( r^2 \)) of the variability in y can be explained by the linear relationship with x.