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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.

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.

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
.

1 answer

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

  1. Calculate P(A):

    • Total students who like apples = 274
    • Total students = 300

    \[ P(A) = \frac{274}{300} \approx 0.9133 \]

  2. Calculate P(B):

    • Total middle school students = 100

    \[ P(B) = \frac{100}{300} = \frac{1}{3} \approx 0.3333 \]

  3. Calculate P(A ∩ B):

    • Students who are in middle school and like apples = 92

    \[ P(A ∩ B) = \frac{92}{300} \approx 0.3067 \]

  4. Check Independence: For A and B to be independent, the following must hold: \[ P(A | B) = P(A) \quad \text{or} \quad P(A ∩ B) = P(A) \cdot P(B) \]

    First, we calculate \( P(A | B) \): \[ P(A | B) = \frac{P(A ∩ B)}{P(B)} = \frac{92/300}{100/300} = \frac{92}{100} = 0.92 \]

  5. Calculate \( P(A) \times P(B) \): \[ P(A) \times P(B) \approx 0.9133 \times \frac{1}{3} \approx 0.3044 \]

  6. Compare P(A ∩ B) and P(A) \cdot P(B):

    • We found \( P(A ∩ B) \approx 0.3067 \)
    • And \( P(A) \cdot P(B) \approx 0.3044 \)

Since \( P(A ∩ B) \neq P(A) \cdot P(B) \) and \( P(A | B) \) is not equal to \( P(A) \), we conclude that the events are dependent.

Thus, the correct response is: No; the events are dependent because P(A)=0.913 and P(A|B)=0.92.