Question
Find the correlation coefficient of the line for these points. Explain
(-3,-40) (1,12) (5,72) and (7, 137),
Since, the equation of a line is,
y = ax+ b
y=17x+3. Please break this down
(-3,-40) (1,12) (5,72) and (7, 137),
Since, the equation of a line is,
y = ax+ b
y=17x+3. Please break this down
Answers
GPT-4o mini
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:
1. Calculate the necessary sums for \( x \), \( y \), \( xy \), \( x^2 \), and \( y^2 \).
2. Use those sums to compute the correlation coefficient using the formula.
3. Verify the points and the linear equation \( y = 17x + 3 \).
### Given points:
1. (-3, -40)
2. (1, 12)
3. (5, 72)
4. (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:
1. For \( x = -3 \): \( y = 17(-3) + 3 = -51 + 3 = -48 \) (Not a point)
2. For \( x = 1 \): \( y = 17(1) + 3 = 17 + 3 = 20 \) (Not a point)
3. For \( x = 5 \): \( y = 17(5) + 3 = 85 + 3 = 88 \) (Not a point)
4. 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.
The steps we will take to find the correlation coefficient include:
1. Calculate the necessary sums for \( x \), \( y \), \( xy \), \( x^2 \), and \( y^2 \).
2. Use those sums to compute the correlation coefficient using the formula.
3. Verify the points and the linear equation \( y = 17x + 3 \).
### Given points:
1. (-3, -40)
2. (1, 12)
3. (5, 72)
4. (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:
1. For \( x = -3 \): \( y = 17(-3) + 3 = -51 + 3 = -48 \) (Not a point)
2. For \( x = 1 \): \( y = 17(1) + 3 = 17 + 3 = 20 \) (Not a point)
3. For \( x = 5 \): \( y = 17(5) + 3 = 85 + 3 = 88 \) (Not a point)
4. 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.