To determine the [tex]\( r \)[/tex]-value, which is the correlation coefficient, we can follow these general steps:
1. Calculate the mean of each variable:
[tex]\[
\bar{x} = \frac{4 + 5 + 8 + 9 + 13}{5} = 39 / 5 = 7.8
\][/tex]
[tex]\[
\bar{y} = \frac{2 + 9 + 10 + 12 + 23}{5} = 56 / 5 = 11.2
\][/tex]
2. Calculate the deviations from the mean for each variable:
[tex]\[
(x_i - \bar{x}): \quad 4 - 7.8, \quad 5 - 7.8, \quad 8 - 7.8, \quad 9 - 7.8, \quad 13 - 7.8
\][/tex]
[tex]\[
(y_i - \bar{y}): \quad 2 - 11.2, \quad 9 - 11.2, \quad 10 - 11.2, \quad 12 - 11.2, \quad 23 - 11.2
\][/tex]
3. Compute the products of the deviations for each pair [tex]\((x_i, y_i)\)[/tex]:
[tex]\[
(x_i - \bar{x}) \cdot (y_i - \bar{y})
\][/tex]
4. Sum up these products:
[tex]\[
\sum (x_i - \bar{x}) \cdot (y_i - \bar{y})
\][/tex]
5. Calculate the squared deviations for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] respectively:
[tex]\[
\sum (x_i - \bar{x})^2
\][/tex]
[tex]\[
\sum (y_i - \bar{y})^2
\][/tex]
6. Plug these into the correlation formula:
[tex]\[
r = \frac{\sum (x_i - \bar{x}) (y_i - \bar{y})}{ \sqrt{\sum (x_i - \bar{x})^2} \sqrt{\sum (y_i - \bar{y})^2} }
\][/tex]
After computing the above steps, the value obtained is:
[tex]\[
r \approx 0.953
\][/tex]
Thus, the [tex]\( r \)[/tex]-value is [tex]\( \boxed{0.953} \)[/tex].
In conclusion, the correct answer is [tex]\( D. \, 0.953 \)[/tex].
