To determine whether 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 probability of being from Texas given that the person prefers brand A [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex].
1. Calculate [tex]\( P(\text{Texas}) \)[/tex]:
The total number of respondents is 275. Out of these, 125 are from Texas.
[tex]\[
P(\text{Texas}) = \frac{\text{Number of respondents from Texas}}{\text{Total number of respondents}} = \frac{125}{275} \approx 0.4545
\][/tex]
2. Calculate [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]:
To find [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex], we look at the number of people who prefer brand A, which is 176, and out of those, the number of people from Texas who prefer brand A, which is 80.
[tex]\[
P(\text{Texas} \mid \text{Brand A}) = \frac{\text{Number of Texas respondents who prefer Brand A}}{\text{Total number of respondents who prefer Brand A}} = \frac{80}{176} \approx 0.4545
\][/tex]
3. Compare the probabilities:
[tex]\[
P(\text{Texas}) \approx 0.4545
\][/tex]
[tex]\[
P(\text{Texas} \mid \text{Brand A}) \approx 0.4545
\][/tex]
Since [tex]\( P(\text{Texas}) \approx P(\text{Texas} \mid \text{Brand A}) \)[/tex], being from Texas and preferring brand A are independent events. Therefore, the correct answer is:
C. 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].
