To determine the value of \( r^2 \) given the independent-measures \( t \) statistic and the sample size, we will go through the following steps:
1. Determine the degrees of freedom (df):
For independent-measures \( t \)-tests, the degrees of freedom are calculated based on the sample size minus one for each group. Since there are two samples, the total degrees of freedom is:
[tex]\[
df = (n - 1) + (n - 1) = 2(n - 1)
\][/tex]
Given \( n = 9 \), we have:
[tex]\[
df = 2(9 - 1) = 2 \cdot 8 = 16
\][/tex]
2. Calculate \( r^2 \):
The effect size \( r^2 \) is calculated using the formula:
[tex]\[
r^2 = \frac{t^2}{t^2 + df}
\][/tex]
Given \( t = 2.00 \) and \( df = 16 \), we can substitute these values into the formula:
[tex]\[
r^2 = \frac{2.00^2}{2.00^2 + 16}
\][/tex]
Calculate \( t^2 \):
[tex]\[
2.00^2 = 4
\][/tex]
Substitute back into the formula:
[tex]\[
r^2 = \frac{4}{4 + 16} = \frac{4}{20}
\][/tex]
3. Compare with the given choices:
From the answer choices provided:
[tex]\[
A. \frac{4}{16}
\][/tex]
[tex]\[
B. \frac{4}{20}
\][/tex]
[tex]\[
C. \frac{2}{16}
\][/tex]
[tex]\[
D. \frac{2}{18}
\][/tex]
You can see that the correct match for the computed \( r^2 = \frac{4}{20} \).
Therefore, the correct answer is [tex]\( B \)[/tex].
