Question

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)=0.90.
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.

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)=0.913 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.

Answer 1

To determine whether events A (a student likes apples) and B (a student is in middle school) are independent, we need to check if the equation \( P(A) = P(A | B) \) holds true. If it does, the events are independent; if not, they are dependent.

1. **Calculate \( P(A) \)**:
- Total students = 300
- Students who like apples = 274
- \( P(A) = \frac{274}{300} = 0.9133 \) (approx 0.913)

2. **Calculate \( P(B) \)**:
- Total middle school students = 100
- \( P(B) = \frac{100}{300} = \frac{1}{3} \) (approx 0.333)

3. **Calculate \( P(A \cap B) \)**:
- Students who like apples and are in middle school = 92
- \( P(A \cap B) = \frac{92}{300} \approx 0.3067 \)

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

Now we can compare \( P(A) \) and \( P(A | B) \):
- \( P(A) \approx 0.913 \)
- \( P(A | B) = 0.92 \)

Since \( P(A) \) (0.913) is not equal to \( P(A | B) \) (0.92), events A and B are not independent; they are dependent.

Based on this analysis, the correct response is:

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