Let's analyze the given table and the possible values for [tex]\( R \)[/tex].
Here is the provided table rewritten for clarity:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & A & B & C & \text{Total} \\
\hline
D & 0.12 & 0.78 & 0.10 & 1.0 \\
\hline
E & R & S & T & 1.0 \\
\hline
\text{Total} & U & X & Y & 1.0 \\
\hline
\end{array}
\][/tex]
First, let's find the missing column totals [tex]\( U \)[/tex], [tex]\( X \)[/tex], and [tex]\( Y \)[/tex]:
- The total for column [tex]\( A \)[/tex] (denoted as [tex]\( U \)[/tex]):
[tex]\[
U = 0.12 + R
\][/tex]
- The total for column [tex]\( B \)[/tex] (denoted as [tex]\( X \)[/tex]):
[tex]\[
X = 0.78 + S
\][/tex]
- The total for column [tex]\( C \)[/tex] (denoted as [tex]\( Y \)[/tex]):
[tex]\[
Y = 0.10 + T
\][/tex]
Given that the overall totals of each column should sum to 1, we have:
- For column [tex]\( A \)[/tex]:
[tex]\[
0.12 + R = 1.0 \implies R = 1.0 - 0.12 = 0.88
\][/tex]
- For column [tex]\( B \)[/tex]:
[tex]\[
0.78 + S = 1.0 \implies S = 1.0 - 0.78 = 0.22
\][/tex]
- For column [tex]\( C \)[/tex]:
[tex]\[
0.10 + T = 1.0 \implies T = 1.0 - 0.10 = 0.90
\][/tex]
From this analysis, let's examine the proposed values for [tex]\( R \)[/tex]:
- [tex]\( 0.09 \)[/tex]
- [tex]\( 0.10 \)[/tex]
- [tex]\( 0.13 \)[/tex]
- [tex]\( 0.79 \)[/tex]
We need to identify which value for [tex]\( R \)[/tex] would indicate an association. Note that in conditional probability and association analysis, large discrepancies between expected and observed frequencies can indicate such associations.
Given that [tex]\( D \)[/tex] represents 78% in column [tex]\( B \)[/tex], we need a value for [tex]\( R \)[/tex] which can practically differentiate column [tex]\( A \)[/tex]. Here, a value like [tex]\( 0.79 \)[/tex] for [tex]\( R \)[/tex] would show that most values of [tex]\( E \)[/tex] are conversing significantly to column [tex]\( A \)[/tex]. It highlights a much higher discrepancy compared to the other proposed values (0.09, 0.10, and 0.13).
The value [tex]\( 0.79 \)[/tex] is much larger, indicating a much stronger association between the variables compared to other provided values which are significantly smaller.
Therefore, the value for [tex]\( R \)[/tex] that most likely indicates an association between the conditional variables is:
[tex]\[
\boxed{0.79}
\][/tex]
