To find the correlation coefficient (r) for the data provided, we first need to organize the data and perform calculations. Here’s the given data in a structured format:
x y
3 7
3 9
4 6
6 11
7 10
8 15
8 16
9 14
10 12
Now, let's calculate the correlation coefficient \( r \).
-
Calculate the means of x and y:
\[ \bar{x} = \frac{3 + 3 + 4 + 6 + 7 + 8 + 8 + 9 + 10}{9} = \frac{58}{9} \approx 6.44 \]
\[ \bar{y} = \frac{7 + 9 + 6 + 11 + 10 + 15 + 16 + 14 + 12}{9} = \frac{99}{9} = 11 \]
-
Calculate the standard deviations of x and y:
Standard deviation formula:
\[ s_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \]
\[ s_y = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n - 1}} \]
After performing the calculations,
-
Calculate the covariance:
\[ \text{cov}(x, y) = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n - 1} \]
-
Calculate the correlation coefficient:
\[ r = \frac{\text{cov}(x, y)}{s_x s_y} \]
After running the calculations, we find that the correlation coefficient \( r \approx 0.79 \).
Thus, the correct answer is:
r = 0.79