Use the table to answer the question. 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 Total 100 100 100 300 Let A be the event that a student likes apples. Let B be the event that a student is in middle school. Based on the data in the table, are events A and B independent? (1 point) Responses Yes; the events are independent because P(A)=P(A∩B)=0.92 . Yes; the events are independent because P(A)=P(A∩B)=0.92 . No; the events are dependent because P(A)=0.93 and P(A∩B)=0.91 . No; the events are dependent because cap p times cap A is equal to 0 point 9 3 and P(A∩B)=0.91 . Yes; the events are independent because P(A)=P(A|B)=0.90 . Yes; the events are independent because cap p times cap A is equal to cap p times open paren cap A divides cap b close paren is equal to 0 point 9 0. No; the events are dependent because P(A)=0.913 and P(A|B)=0.92 .

To determine whether events A (a student likes apples) and B (a student is in middle school) are independent, we will follow these steps:

1. Find \( P(A) \), the overall probability that a student likes apples.
2. Find \( P(B) \), the probability that a student is in middle school.
3. Find \( P(A \cap B) \), the probability that a student is in middle school and likes apples.
4. Check if \( P(A \cap B) = P(A) \times P(B) \). If this holds true, then A and B are independent.

From the data in the table:

• Total number of students = 300
• Students who like apples (Total) = 274

Thus,

\[ P(A) = \frac{\text{Number of students who like apples}}{\text{Total number of students}} = \frac{274}{300} \approx 0.9133 \]

Now, find \( P(B) \):

• Total middle school students = 100

Thus,

\[ P(B) = \frac{\text{Number of middle school students}}{\text{Total number of students}} = \frac{100}{300} = \frac{1}{3} \approx 0.3333 \]

Now, find \( P(A \cap B) \):

• Students in middle school who like apples = 92

Thus,

\[ P(A \cap B) = \frac{\text{Number of middle school students who like apples}}{\text{Total number of students}} = \frac{92}{300} \approx 0.3067 \]

Next, calculate \( P(A) \times P(B) \):

\[ P(A) \times P(B) = 0.9133 \times 0.3333 \approx 0.3044 \]

Finally, compare \( P(A \cap B) \) and \( P(A) \times P(B) \):

• \( P(A \cap B) \approx 0.3067 \)
• \( P(A) \times P(B) \approx 0.3044 \)

Since \( P(A \cap B) \neq P(A) \times P(B) \), events A and B are dependent.

Thus, the correct answer is:

No; the events are dependent because \( P(A) \approx 0.9133 \) and \( P(A \cap B) \approx 0.3067 \).