To determine if the events "being from Texas" and "preferring brand A" are independent, we need to compare the probability of being from Texas [tex]\( P(\text{Texas}) \)[/tex] with the conditional probability of being from Texas given preferring brand A [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]. If these two probabilities are equal, the events are independent.
Here are the steps to find these probabilities and determine if the events are independent:
### Step 1: Calculate the Probability of Being from Texas [tex]\( P(\text{Texas}) \)[/tex]
The total number of participants is 275.
The number of participants from Texas is 125.
[tex]\[ P(\text{Texas}) = \frac{\text{Number of participants from Texas}}{\text{Total number of participants}} = \frac{125}{275} \approx 0.4545 \][/tex]
### Step 2: Calculate the Probability of Preferring Brand A [tex]\( P(\text{Brand A}) \)[/tex]
The number of participants who prefer Brand A is 176.
[tex]\[ P(\text{Brand A}) = \frac{\text{Number of participants who prefer Brand A}}{\text{Total number of participants}} = \frac{176}{275} \approx 0.64 \][/tex]
### Step 3: Calculate the Conditional Probability of Being from Texas Given Preferring Brand A [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]
The number of participants from Texas who prefer Brand A is 80.
The total number of participants who prefer Brand A is 176.
[tex]\[ P(\text{Texas} \mid \text{Brand A}) = \frac{\text{Number of participants from Texas who prefer Brand A}}{\text{Total number of participants who prefer Brand A}} = \frac{80}{176} \approx 0.4545 \][/tex]
### Step 4: Compare Probabilities to Determine Independence
Compare [tex]\( P(\text{Texas}) \)[/tex] and [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]:
[tex]\[ P(\text{Texas}) = 0.4545 \][/tex]
[tex]\[ P(\text{Texas} \mid \text{Brand A}) = 0.4545 \][/tex]
Since [tex]\( P(\text{Texas}) \approx P(\text{Texas} \mid \text{Brand A}) \)[/tex], the events "being from Texas" and "preferring brand A" are independent.
### Conclusion
The correct answer is:
A. No, they are not independent because [tex]\( P(\text{Texas}) \approx 0.45 \)[/tex] and [tex]\( P(\text{Texas} \mid \text{Brand A}) \approx 0.45 \)[/tex].
