To determine whether being from California and preferring brand [tex]$A$[/tex] are independent events, we need to compare two probabilities:
1. [tex]\( P(\text{California}) \)[/tex]: The probability that a randomly selected person is from California.
2. [tex]\( P(\text{California} \mid \text{brand A}) \)[/tex]: The probability that a person is from California given that they prefer brand [tex]$A$[/tex].
If these two probabilities are equal, the events are independent. Otherwise, they are not.
### Step-by-Step Solution:
1. Calculate [tex]\( P(\text{California}) \)[/tex]:
[tex]\[ P(\text{California}) = \frac{\text{Number of California people}}{\text{Total number of people}} = \frac{150}{275} \approx 0.5454545454545454 \][/tex]
2. Calculate [tex]\( P(\text{California} \mid \text{brand A}) \)[/tex]:
[tex]\[ P(\text{California} \mid \text{brand A}) = \frac{\text{Number of people from California who prefer brand A}}{\text{Total number of people who prefer brand A}} = \frac{96}{176} \approx 0.5454545454545454 \][/tex]
3. Compare the probabilities:
[tex]\[ P(\text{California}) \approx 0.55 \][/tex]
[tex]\[ P(\text{California} \mid \text{brand A}) \approx 0.55 \][/tex]
Since [tex]\( P(\text{California}) \)[/tex] is approximately equal to [tex]\( P(\text{California} \mid \text{brand A}) \)[/tex], the two events (being from California and preferring brand A) are independent.
### Conclusion:
The correct answer is:
B. Yes, they are independent because [tex]\( P(\text{California}) \approx 0.55 \)[/tex] and [tex]\( P(\text{California} \mid \text{brand A}) \approx 0.55 \)[/tex].
