To determine whether being from California and preferring brand B are independent events, we need to check if the probability of being from California and preferring brand B equals the product of the individual probabilities of being from California and preferring brand B. Let us break it down step by step:
1. Calculate [tex]\( P(\text{California}) \)[/tex]:
- The total number of people surveyed is 275.
- The number of people from California is 150.
- Therefore, [tex]\( P(\text{California}) = \frac{150}{275} \approx 0.545 \)[/tex].
2. Calculate [tex]\( P(\text{Brand B}) \)[/tex]:
- The total number of people who prefer Brand B is 99.
- Therefore, [tex]\( P(\text{Brand B}) = \frac{99}{275} \approx 0.36 \)[/tex].
3. Calculate [tex]\( P(\text{California and Brand B}) \)[/tex]:
- The number of people from California who prefer Brand B is 54.
- Therefore, [tex]\( P(\text{California and Brand B}) = \frac{54}{275} \approx 0.196 \)[/tex].
4. Calculate [tex]\( P(\text{California} | \text{Brand B}) \)[/tex]:
- The number of people who prefer Brand B is 99.
- The number of those who are from California and prefer Brand B is 54.
- Therefore, [tex]\( P(\text{California} | \text{Brand B}) = \frac{54}{99} \approx 0.545 \)[/tex].
5. Check for Independence:
- Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if [tex]\( P(A \text{ and } B) = P(A) \cdot P(B) \)[/tex].
- Calculate [tex]\( P(\text{California}) \times P(\text{Brand B}) \)[/tex]:
[tex]\[
P(\text{California}) \times P(\text{Brand B}) = 0.545 \times 0.36 \approx 0.196.
\][/tex]
- Compare this with [tex]\( P(\text{California and Brand B}) \)[/tex]:
[tex]\[
P(\text{California and Brand B}) \approx 0.196.
\][/tex]
- Since [tex]\( P(\text{California}) \times P(\text{Brand B}) \approx P(\text{California and Brand B}) \)[/tex], the events are independent.
Therefore, the correct option is:
A. Yes, they are independent because [tex]\( P(\text{California}) \approx 0.55 \)[/tex] and [tex]\( P(\text{California} | \text{Brand B}) \approx 0.55 \)[/tex].
