Use the table to answer the question.

Apple Preferences of Elementary, Middle, and High School Students
Grade Level

Apple Preference Elementary School Students Middle School Students High School Students Total

Likes apples
85
92
97
274
Dislikes apples
15
8
3
26
(1 point)
• No: the events are dependent because P(A) - 0.913 and P(A|B) = 0.92
• No, the events are dependent because P(A) — 0.93 and P(A n B) = 0.91.
O Yes, the events are independent because P(A) - P(A B) — 0.90.
• Yes: the events are independent because P(A) - P(An B) = 0.92

1 answer

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.

  1. 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)

  1. 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.

  2. 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.