Question
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)=0.92
.
Yes; the events are independent because P(A)=P(A∩B)=0.92
.
Yes; the events are independent because P(A)=P(A|B)=0.90
.
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)=0.93
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
.
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.
Skip to navigation
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
.
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)=0.93
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
.
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.
Skip to navigation
Answers
Answered by
GPT-4o mini
To determine whether the events A (a student likes apples) and B (a student is in middle school) are independent, we need to calculate the probabilities involved.
1. **Calculate P(A)**: The probability that a student likes apples.
\[
P(A) = \frac{\text{Total number of students who like apples}}{\text{Total number of students}} = \frac{274}{300} \approx 0.9133
\]
2. **Calculate P(B)**: The probability that a student is in middle school.
\[
P(B) = \frac{\text{Total number of middle school students}}{\text{Total number of students}} = \frac{100}{300} = \frac{1}{3} \approx 0.3333
\]
3. **Calculate P(A ∩ B)**: The probability that a student likes apples and is in middle school.
\[
P(A \cap B) = \frac{\text{Number of middle school students who like apples}}{\text{Total number of students}} = \frac{92}{300} \approx 0.3067
\]
4. **Check independence**: Events A and B are independent if \( P(A \cap B) = P(A) \cdot P(B) \).
Let's calculate \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) \approx 0.9133 \cdot 0.3333 \approx 0.3044
\]
Since \( P(A \cap B) \approx 0.3067 \) is not equal to \( P(A) \cdot P(B) \approx 0.3044 \), the events A and B are dependent.
Therefore, the correct answer is:
**No; the events are dependent because P(A)=0.9133 and P(A ∩ B)=0.3067.**
1. **Calculate P(A)**: The probability that a student likes apples.
\[
P(A) = \frac{\text{Total number of students who like apples}}{\text{Total number of students}} = \frac{274}{300} \approx 0.9133
\]
2. **Calculate P(B)**: The probability that a student is in middle school.
\[
P(B) = \frac{\text{Total number of middle school students}}{\text{Total number of students}} = \frac{100}{300} = \frac{1}{3} \approx 0.3333
\]
3. **Calculate P(A ∩ B)**: The probability that a student likes apples and is in middle school.
\[
P(A \cap B) = \frac{\text{Number of middle school students who like apples}}{\text{Total number of students}} = \frac{92}{300} \approx 0.3067
\]
4. **Check independence**: Events A and B are independent if \( P(A \cap B) = P(A) \cdot P(B) \).
Let's calculate \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) \approx 0.9133 \cdot 0.3333 \approx 0.3044
\]
Since \( P(A \cap B) \approx 0.3067 \) is not equal to \( P(A) \cdot P(B) \approx 0.3044 \), the events A and B are dependent.
Therefore, the correct answer is:
**No; the events are dependent because P(A)=0.9133 and P(A ∩ B)=0.3067.**
There are no human answers yet. A form for humans to post answers is coming very soon!