To find the correlation coefficient \( r \) for the given points and the line, we first need to understand what the correlation coefficient represents. It quantifies the degree to which two variables (in this case, \( x \) and \( y \)) are linearly related.
The steps we will take to find the correlation coefficient include:
- Calculate the necessary sums for \( x \), \( y \), \( xy \), \( x^2 \), and \( y^2 \).
- Use those sums to compute the correlation coefficient using the formula.
- Verify the points and the linear equation \( y = 17x + 3 \).
Given points:
- (-3, -40)
- (1, 12)
- (5, 72)
- (7, 137)
Step 1: Calculate necessary sums
Let's represent the points and calculate all the necessary sums:
\[ \begin{align*} \text{Point} & \quad x & \quad y & \quad xy & \quad x^2 & \quad y^2 \ (-3, -40) & \quad -3 & -40 & 120 & 9 & 1600 \ (1, 12) & \quad 1 & 12 & 12 & 1 & 144 \ (5, 72) & \quad 5 & 72 & 360 & 25 & 5184 \ (7, 137) & \quad 7 & 137 & 959 & 49 & 18769 \ \end{align*} \]
Now we can calculate the sums:
\[ \begin{align*} \sum x &= -3 + 1 + 5 + 7 = 10 \ \sum y &= -40 + 12 + 72 + 137 = 181 \ \sum xy &= 120 + 12 + 360 + 959 = 1451 \ \sum x^2 &= 9 + 1 + 25 + 49 = 84 \ \sum y^2 &= 1600 + 144 + 5184 + 18769 = 26097 \ n &= 4 \quad \text{(number of points)} \ \end{align*} \]
Step 2: Calculate the correlation coefficient
The formula for the correlation coefficient \( r \) is given by:
\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]
Substituting the values we calculated:
\[ r = \frac{4(1451) - (10)(181)}{\sqrt{[4(84) - (10)^2][4(26097) - (181)^2]}} \]
Calculating the numerator:
\[ = \frac{5804 - 1810}{\sqrt{[336 - 100][104388 - 32761]}} = \frac{3994}{\sqrt{236 \cdot 71627}} \]
Calculating \( 236 \cdot 71627 \):
\[ = 16873132 \]
Taking the square root:
\[ \sqrt{16873132} \approx 4107.59 \]
Now, substituting back into the equation for \( r \):
\[ r \approx \frac{3994}{4107.59} \approx 0.972 \]
Conclusion
The correlation coefficient \( r \) for the given points is approximately 0.972, indicating a strong positive linear relationship between the variables \( x \) and \( y \).
Equation Verification
You provided the linear equation \( y = 17x + 3 \). We can check if these points lie on the specified line by plugging in the \( x \) values:
- For \( x = -3 \): \( y = 17(-3) + 3 = -51 + 3 = -48 \) (Not a point)
- For \( x = 1 \): \( y = 17(1) + 3 = 17 + 3 = 20 \) (Not a point)
- For \( x = 5 \): \( y = 17(5) + 3 = 85 + 3 = 88 \) (Not a point)
- For \( x = 7 \): \( y = 17(7) + 3 = 119 + 3 = 122 \) (Not a point)
This indicates that the points provided do not actually lie on the line defined by \( y = 17x + 3 \). You may need to verify the linear relationship or recheck the equation based on the points provided in your query.