To determine which value for [tex]\( R \)[/tex] would most likely indicate an association between the conditional variables in the given table, we need to carefully examine the relationships between the known and unknown quantities across the rows and columns. Here is a step-by-step approach:
1. Understand Given Values:
- From the table, we have the first row corresponding to D:
[tex]\[ D_A = 0.12, \quad D_B = 0.78, \quad D_C = 0.10, \quad \text{and} \quad D_{\text{Total}} = 1.0 \][/tex]
2. E (Second Row) Values:
- The second row sums up to 1.0:
[tex]\[ R + S + T = 1.0 \][/tex]
3. Total (Third Row) Values:
- The totals for each column are represented as:
[tex]\[ U, \quad X, \quad Y, \quad \text{and} \quad 1.0 \][/tex]
4. Compute Total Column Values:
- From D and E rows, each total column value is derived by summing the corresponding D and E values:
[tex]\[ U = D_A + R \][/tex]
[tex]\[ X = D_B + S \][/tex]
[tex]\[ Y = D_C + T \][/tex]
5. Relationships to Maintain Total Sum:
- By substituting the known values and the constraint that the totals must sum to 1.0, we derive:
[tex]\[ U = 0.12 + R \][/tex]
[tex]\[ X = 0.78 + S \][/tex]
[tex]\[ Y = 0.10 + T \][/tex]
6. Determine Consistent Values:
- Substitute the possible values for [tex]\( R \)[/tex] — 0.09, 0.10, 0.13, 0.79 — into the total equations and check which value results in consistent associations:
- For [tex]\( R = 0.09 \)[/tex]:
[tex]\[ S + T = 1.0 - 0.09 = 0.91 \][/tex]
- Substitute and solve:
[tex]\[ U = 0.12 + 0.09 = 0.21 \][/tex]
[tex]\[ Y = 0.10 + T \][/tex]
[tex]\[ T = Y - 0.10 \][/tex]
- Substitute values into S calculation:
[tex]\[ 0.78 + S + (Y - 0.10) = 1.0 - 0.21 \][/tex]
- Analyze for column consistency doesn't meet our total column values.
- For [tex]\( R = 0.10 \)[/tex]:
[tex]\[ S + T = 1.0 - 0.10 = 0.90 \][/tex]
- Analogous check for consistent total values doesn't solve our constraints.
- For [tex]\( R = 0.13 \)[/tex]:
[tex]\[ S + T = 1.0 - 0.13 = 0.87 \][/tex]
- Similar steps to find mismatch on total compared to others.
- For [tex]\( R = 0.79 \)[/tex]:
[tex]\[ S + T = 1.0 - 0.79 = 0.21 \][/tex]
- Assess correctness:
[tex]\[ U = 0.12 + 0.79 = 0.91 \][/tex]
[tex]\[ 0.78 + (1.0 - 0.10) - 0.79 - 0.1 + 0.79\][/tex]
[tex]\[ 0.78 + 0.21 = 1.0 \right]
\[ consistent D associations correctly returning consistent valid total sum. ]
Therefore, the value for \( R \) that will most likely indicate an association between the conditional variables is:
\[ R = 0.79 \][/tex]
