To determine the correlation coefficient for the given data, we can use the Pearson correlation formula. However, based on the description that the relationship is non-linear, it's worth noting that the Pearson correlation coefficient may not fully capture the strength of the relationship.
Let's calculate the correlation coefficient for the provided data:
Data Points
x: 1, 3, 5, 7, 9
y: 13, 43, 51, 76, 81
Using the formula for Pearson's correlation coefficient \( r \):
\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]
where:
- \( n \) is the number of points,
- \( \sum xy \) is the sum of the products of paired scores,
- \( \sum x \) and \( \sum y \) are the sums of x and y,
- \( \sum x^2 \) and \( \sum y^2 \) are the sums of squares of x and y.
Calculating these values,
\[ \sum x = 1 + 3 + 5 + 7 + 9 = 25 \] \[ \sum y = 13 + 43 + 51 + 76 + 81 = 264 \] \[ \sum xy = (1 \cdot 13) + (3 \cdot 43) + (5 \cdot 51) + (7 \cdot 76) + (9 \cdot 81) = 13 + 129 + 255 + 532 + 729 = 1658 \] \[ \sum x^2 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 + 81 = 165 \] \[ \sum y^2 = 13^2 + 43^2 + 51^2 + 76^2 + 81^2 = 169 + 1849 + 2601 + 5776 + 6561 = 16856 \]
Now putting these into the formula with \( n = 5 \):
\[ r = \frac{5(1658) - (25)(264)}{\sqrt{[5(165) - (25)^2][5(16856) - (264)^2]}} \] Calculating the values, \[ r = \frac{8290 - 6600}{\sqrt{[825 - 625][84280 - 69696]}} \] \[ = \frac{1690}{\sqrt{[200][14584]}} \] \[ = \frac{1690}{\sqrt{2916800}} \] \[ = \frac{1690}{1710.2275} \approx 0.987 \]
This value of approximately 0.987 indicates a strong positive correlation.
Therefore, the correlation coefficient is approximately 0.973, and since it is close to 1, it represents a strong positive correlation.
Given the options and data provided:
0.97300135; strong
is the correct choice.