Use the table to answer the question.

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)=0.90.
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.
No; the events are dependent because cap p times cap A is equal to 0 point 9 1 3 and cap p times open paren cap A divides cap b close paren is equal to 0 point 9 2.

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)=0.92.

Answer 1

To determine if events A (a student likes apples) and B (a student is in middle school) are independent, we need to calculate the probabilities P(A), P(B), and P(A|B).

1. Calculate P(A) (the probability that a student likes apples): \[ P(A) = \frac{\text{Total who like apples}}{\text{Total students}} = \frac{274}{300} \approx 0.913 \]

2. Calculate P(B) (the probability that a student is in middle school): \[ P(B) = \frac{\text{Total middle school students}}{\text{Total students}} = \frac{100}{300} = \frac{1}{3} \approx 0.333 \]

3. Calculate P(A ∩ B) (the probability that a student likes apples and is in middle school): \[ P(A \cap B) = \frac{\text{Middle school students who like apples}}{\text{Total students}} = \frac{92}{300} \approx 0.307 \]

4. Calculate P(A|B) (the probability that a student likes apples given that the student is in middle school): \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \approx \frac{92/300}{100/300} = \frac{92}{100} = 0.92 \]

Now, check if \( P(A) \) is equal to \( P(A|B) \):

• \( P(A) \approx 0.913 \)
• \( P(A|B) = 0.92 \)

Since \( P(A) \) is not equal to \( P(A|B) \), events A and B are dependent.

Therefore, the correct response is:

No; the events are dependent because P(A)=0.913 and P(A|B)=0.92.