To determine the independence of the events based on the table provided, we can calculate the probabilities and check if they follow the condition for independence.
- Calculate P(A): This is the probability that a student likes apples.
\( P(A) = \frac{\text{Total likes apples}}{\text{Total students}} = \frac{274}{300} = 0.9133 \) (since 274 is the total who like apples and 300 is the total number of students = 274 likes + 26 dislikes)
-
Calculate P(B): This is the probability of being from a particular grade level. Since the table does not provide specifics about the grade levels, we will need to assume or calculate based on overall distributions.
-
Calculate P(A ∩ B): This is the combined probability for both liking apples and being from a particular grade level, which remains unspecified.
Now, let's simplify whether the conclusions drawn in the answer choices can be confirmed:
- If P(A|B) denotes the probability of liking apples given that the grade level (let’s say middle school) is specified, then you can calculate this with given data.
- Middle School: Likes = 92, Total = 92 + 8 = 100; thus, \( P(A|B) = \frac{92}{100} = 0.92 \)
To conclude whether the events are independent, we check if:
P(A|B) = P(A).
- We found that \( P(A) \approx 0.9133 \) and \( P(A|B) = 0.92 \) for the middle school group.
Since \( P(A|B) \) is not equal to \( P(A) \), the events are dependent.
Based on this analysis, the correct answer is:
- No: the events are dependent because P(A) = 0.913 and P(A|B) = 0.92.
So, the answer to the question based on the interpretation of the options is:
- No: the events are dependent because P(A) = 0.913 and P(A|B) = 0.92.