Question

What is the correlation coefficient of the following data? Is it a strong or weak correlation?

x 1 3 5 7 9
y 13 43 51 76 81
(1 point)
Responses

0.97300135; weak
0.97300135; weak

−0.97300135
; weak
negative 0.97300135 ; weak

0.97300135; strong
0.97300135; strong

−0.97300135
; strong

Answer 1

To calculate the correlation coefficient (Pearson's r) for the given data, you can use the formula:

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

Where:
- \(n\) is the number of data points,
- \(\sum xy\) is the sum of the product of x and y,
- \(\sum x\) is the sum of x values,
- \(\sum y\) is the sum of y values,
- \(\sum x^2\) is the sum of x squared values,
- \(\sum y^2\) is the sum of y squared values.

First, we calculate the necessary sums:

1. For \(x\):
- \(\sum x = 1 + 3 + 5 + 7 + 9 = 25\)
- \(\sum x^2 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 + 81 = 165\)

2. For \(y\):
- \(\sum y = 13 + 43 + 51 + 76 + 81 = 264\)
- \(\sum y^2 = 13^2 + 43^2 + 51^2 + 76^2 + 81^2 = 169 + 1849 + 2601 + 5776 + 6561 = 16856\)

3. For \(xy\):
- \(xy = 1 \cdot 13 + 3 \cdot 43 + 5 \cdot 51 + 7 \cdot 76 + 9 \cdot 81 = 13 + 129 + 255 + 532 + 729 = 1658\)

With these sums:
- \(n = 5\)
- \(\sum xy = 1658\)
- \(\sum x = 25\)
- \(\sum y = 264\)
- \(\sum x^2 = 165\)
- \(\sum y^2 = 16856\)

Now plug in the values into the formula for \(r\):

\[
r = \frac{5(1658) - (25)(264)}{\sqrt{[5(165) - (25)^2][5(16856) - (264)^2]}}
\]

Calculating the numerator:

\[
5(1658) = 8290
\]
\[
(25)(264) = 6600
\]
\[
\text{Numerator} = 8290 - 6600 = 1690
\]

Calculating the denominator:

\[
5(165) - (25)^2 = 825 - 625 = 200
\]
\[
5(16856) - (264)^2 = 84280 - 69696 = 14584
\]
\[
\text{Denominator} = \sqrt{200 \cdot 14584} \approx \sqrt{2916800} \approx 1710.09
\]

Finally, calculating \(r\):

\[
r \approx \frac{1690}{1710.09} \approx 0.986
\]

Given that \(r\) is approximately \(0.986\), it's close to \(1\), indicating a **strong positive correlation**.

Thus, the best answer from the provided options is:
**0.97300135; strong**.