### Step-by-Step Solution:
Given the contingency table:
| | [tex]\(X\)[/tex] | [tex]\(Y\)[/tex] | [tex]\(Z\)[/tex] | Total |
|---|---|---|---|---|
| [tex]\(A\)[/tex] | 10 | 80 | 61 | 151 |
| [tex]\(B\)[/tex] | 110 | 44 | 126 | 280 |
| [tex]\(C\)[/tex] | 60 | 59 | 110 | 229 |
| Total | 180 | 183 | 297 | 660 |
We need to determine if [tex]\(Z\)[/tex] and [tex]\(B\)[/tex] are independent events. To do this, we will check if [tex]\(P(Z \mid B) = P(Z)\)[/tex].
1. Calculate the total number of observations:
[tex]\[
\text{Total observations} = 660
\][/tex]
2. Calculate [tex]\(P(B)\)[/tex]:
- [tex]\(B\)[/tex] is the event where the row is [tex]\(B\)[/tex].
- Total number of [tex]\(B\)[/tex] observations is 280.
[tex]\[
P(B) = \frac{\text{Total } B \text{ observations}}{\text{Total observations}} = \frac{280}{660} \approx 0.42424242424242425
\][/tex]
3. Calculate [tex]\(P(Z)\)[/tex]:
- [tex]\(Z\)[/tex] is the event where the column is [tex]\(Z\)[/tex].
- Total number of [tex]\(Z\)[/tex] observations is 297.
[tex]\[
P(Z) = \frac{\text{Total } Z \text{ observations}}{\text{Total observations}} = \frac{297}{660} \approx 0.45
\][/tex]
4. Calculate [tex]\(P(Z \mid B)\)[/tex]:
- [tex]\(P(Z \mid B)\)[/tex] is the probability of [tex]\(Z\)[/tex] given [tex]\(B\)[/tex].
- Number of observations where both [tex]\(Z\)[/tex] and [tex]\(B\)[/tex] occur is 126.
[tex]\[
P(Z \mid B) = \frac{\text{Number of } (Z \cap B) \text{ observations}}{\text{Total } B \text{ observations}} = \frac{126}{280} \approx 0.45
\][/tex]
5. Check for independence:
- To check if [tex]\(Z\)[/tex] and [tex]\(B\)[/tex] are independent, we need to see if [tex]\(P(Z \mid B) = P(Z)\)[/tex].
[tex]\[
P(Z \mid B) = 0.45
\][/tex]
[tex]\[
P(Z) = 0.45
\][/tex]
Since [tex]\(P(Z \mid B) = P(Z)\)[/tex] is true, [tex]\(Z\)[/tex] and [tex]\(B\)[/tex] are independent events.
### Conclusion:
The statement that is true given the computed probabilities is:
- [tex]\(Z\)[/tex] and [tex]\(B\)[/tex] are independent events because [tex]\(P(Z \mid B) = P(Z)\)[/tex].
