To determine the expected values for the chi-square test of independence, we utilize the formula for expected counts in a contingency table:
[tex]\[ E_{ij} = \frac{( \text{Row Total}_i \times \text{Column Total}_j )}{\text{Grand Total}} \][/tex]
Now, we'll calculate the expected counts for each cell in our table:
- Total group size (Grand Total): 83
- Republican Total: 42
- Democrat Total: 41
- Total in favor: 18
- Total indifferent: 32
- Total opposed: 33
### Republicans:
1. In favor:
[tex]\[
\text{Expected value} = \frac{(42 \times 18)}{83} = 9.108433734939759
\][/tex]
2. Indifferent:
[tex]\[
\text{Expected value} = \frac{(42 \times 32)}{83} = 16.19277108433735
\][/tex]
3. Opposed:
[tex]\[
\text{Expected value} = \frac{(42 \times 33)}{83} = 16.698795180722893
\][/tex]
### Democrats:
1. In favor:
[tex]\[
\text{Expected value} = \frac{(41 \times 18)}{83} = 8.891566265060241
\][/tex]
2. Indifferent:
[tex]\[
\text{Expected value} = \frac{(41 \times 32)}{83} = 15.80722891566265
\][/tex]
3. Opposed:
[tex]\[
\text{Expected value} = \frac{(41 \times 33)}{83} = 16.301204819277107
\][/tex]
Therefore, the table containing the expected values is:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline & \text{In favor} & \text{Indifferent} & \text{Opposed} & \text{Row Total} \\
\hline \text{Republicans} & 9.108 & 16.193 & 16.699 & 42 \\
\hline \text{Democrats} & 8.892 & 15.807 & 16.301 & 41 \\
\hline \text{Column Total} & 18 & 32 & 33 & 83 \\
\hline
\end{array}
\][/tex]
This matches the calculated expected values perfectly. None of the stored options seem to be a perfect numerical match for our calculation, so please verify if there's an error in the options themselves. But based on our complete and accurate calculation, the correct values to present are:
- Republicans: In favor: 9.108 | Indifferent: 16.193 | Opposed: 16.699
- Democrats: In favor: 8.892 | Indifferent: 15.807 | Opposed: 16.301
