To find the [tex]$r$[/tex]-value (correlation coefficient) of the given data, follow these steps:
1. List the data points:
The data, [tex]$(x, y)$[/tex], consists of the pairs:
[tex]\[
(1, 20), (3, 14), (5, 10), (9, 6), (16, 4)
\][/tex]
2. Calculate the means of [tex]$x$[/tex] and [tex]$y$[/tex]:
- The mean of [tex]$x$[/tex], denoted [tex]$\bar{x}$[/tex], is computed as:
[tex]\[
\bar{x} = \frac{1 + 3 + 5 + 9 + 16}{5} = \frac{34}{5} = 6.8
\][/tex]
- The mean of [tex]$y$[/tex], denoted [tex]$\bar{y}$[/tex], is computed as:
[tex]\[
\bar{y} = \frac{20 + 14 + 10 + 6 + 4}{5} = \frac{54}{5} = 10.8
\][/tex]
3. Calculate the deviations from the mean for [tex]$x$[/tex] and [tex]$y$[/tex]:
- For [tex]$x$[/tex]: [tex]$x_i - \bar{x}$[/tex]
- For [tex]$y$[/tex]: [tex]$y_i - \bar{y}$[/tex]
Compute these deviations for each data point.
4. Compute the product of deviations for each pair:
[tex]\[
(x_i - \bar{x})(y_i - \bar{y})
\][/tex]
5. Compute the square of deviations for each [tex]$x$[/tex] and [tex]$y$[/tex]:
[tex]\[
(x_i - \bar{x})^2 \quad \text{and} \quad (y_i - \bar{y})^2
\][/tex]
6. Sum these products and squares:
- Sum of the product of deviations, [tex]$\sum (x_i - \bar{x})(y_i - \bar{y})$[/tex].
- Sum of the square of deviations for [tex]$x$[/tex], [tex]$\sum (x_i - \bar{x})^2$[/tex].
- Sum of the square of deviations for [tex]$y$[/tex], [tex]$\sum (y_i - \bar{y})^2$[/tex].
7. Calculate the correlation coefficient [tex]$r$[/tex]:
[tex]\[
r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}
\][/tex]
8. Round the result to three decimal places.
After performing the above calculations (either manually or using a statistical calculator), we find that the correlation coefficient [tex]$r$[/tex] rounds to:
[tex]\[
\boxed{-0.901}
\][/tex]
So, the answer is A. -0.901.
