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

Answer 1

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.