To determine if being from Texas and preferring Brand A are independent events, we need to compare the probability of being from Texas [tex]\( P(\text{Texas}) \)[/tex] with the conditional probability of being from Texas given that a person prefers Brand A [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex].
First, let's calculate [tex]\( P(\text{Texas}) \)[/tex]:
[tex]\[ P(\text{Texas}) = \frac{\text{Number of people from Texas}}{\text{Total number of people}} = \frac{125}{275} \approx 0.4545 \][/tex]
Next, let's calculate [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]:
[tex]\[ P(\text{Texas} \mid \text{Brand A}) = \frac{\text{Number of Texans preferring Brand A}}{\text{Number of people preferring Brand A}} = \frac{80}{176} \approx 0.4545 \][/tex]
Now, let's compare the probabilities:
- [tex]\( P(\text{Texas}) \approx 0.4545 \)[/tex]
- [tex]\( P(\text{Texas} \mid \text{Brand A}) \approx 0.4545 \)[/tex]
If [tex]\( P(\text{Texas}) \)[/tex] is equal to [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex], it indicates that the events are independent. Since both probabilities are approximately equal, [tex]\( P(\text{Texas}) \approx P(\text{Texas} \mid \text{Brand A}) \)[/tex], we conclude that the events are independent.
Therefore, the answer is:
B. Yes, they are independent because [tex]\( P(\text{Texas}) \approx 0.45 \)[/tex] and [tex]\( P(\text{Texas} \mid \text{Brand A}) \approx 0.45 \)[/tex].
