To determine whether the events "being from California" and "preferring brand A" are independent, we need to compare two probabilities:
1. The probability of being from California, [tex]\( P(\text{California}) \)[/tex].
2. The conditional probability of being from California given that the person prefers brand A, [tex]\( P(\text{California} \mid \text{Brand A}) \)[/tex].
Step 1: Calculate [tex]\( P(\text{California}) \)[/tex]
The total number of people surveyed is 275, and the number of people from California is 150. Therefore, the probability of being from California is:
[tex]\[
P(\text{California}) = \frac{\text{Number of people from California}}{\text{Total number of people}} = \frac{150}{275}
\][/tex]
Simplifying this fraction:
[tex]\[
P(\text{California}) = \frac{150}{275} \approx 0.545
\][/tex]
Step 2: Calculate [tex]\( P(\text{California} \mid \text{Brand A}) \)[/tex]
Next, we need to find the conditional probability that a person is from California given they prefer brand A. This is given by the ratio of the number of people from California who prefer brand A to the total number of people who prefer brand A. According to the table, 96 people from California prefer brand A out of a total of 176 people who prefer brand A.
[tex]\[
P(\text{California} \mid \text{Brand A}) = \frac{\text{Number of people from California who prefer Brand A}}{\text{Total number of people who prefer Brand A}} = \frac{96}{176}
\][/tex]
Simplifying this fraction:
[tex]\[
P(\text{California} \mid \text{Brand A}) = \frac{96}{176} \approx 0.545
\][/tex]
Step 3: Determine Dependence or Independence
For two events to be independent, the probability of one event must be equal to the conditional probability given the other event. In other words, [tex]\( P(\text{California}) \)[/tex] should equal [tex]\( P(\text{California} \mid \text{Brand A}) \)[/tex].
From the calculations:
[tex]\[
P(\text{California}) \approx 0.545
\][/tex]
[tex]\[
P(\text{California} \mid \text{Brand A}) \approx 0.545
\][/tex]
Since these two probabilities are approximately equal, the two events being from California and preferring Brand A are independent.
Therefore, 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].
