1. a survey of students at an art school in kalamazoo, michigan, revealed that 62% prefer classical music…
1. a survey of students at an art school in kalamazoo, michigan, revealed that 62% prefer classical music when studying rather than no music at all. in addition, 37% of students indicated that they are left - handed. of those surveyed, 14% said they are left - handed and do not prefer classical music.\na. complete the two - way relative frequency table.\n| | left - handed (l)| not left - handed (~l)| total|\n|--|--|--|--|\n| prefer classical music (c)| | | 62%|\n| do not prefer classical music (~c)| 14%| | |\n| total| 37%| | |\nb. given that a randomly selected student is not left - handed, what is the probability that they prefer classical music?
Answer
Explanation:
Step1: Find left - handed and prefer classical music
We know that the percentage of left - handed students is 37% and the percentage of left - handed students who do not prefer classical music is 14%. So the percentage of left - handed students who prefer classical music is $37%-14% = 23%$.
Step2: Find not left - handed and prefer classical music
The percentage of students who prefer classical music is 62%, and 23% of those who prefer classical music are left - handed. So the percentage of not left - handed students who prefer classical music is $62%-23%=39%$.
Step3: Find not left - handed and do not prefer classical music
The percentage of students who do not prefer classical music is $100%-62% = 38%$. Since 14% of students are left - handed and do not prefer classical music, the percentage of not left - handed students who do not prefer classical music is $38%-14% = 24%$.
Step4: Find the percentage of not left - handed students
The percentage of left - handed students is 37%, so the percentage of not left - handed students is $100%-37%=63%$.
Step5: Find the total percentages
The total percentage of all students is 100%.
The completed two - way relative frequency table:
| Left - handed ($L$) | Not left - handed ($\sim L$) | Total | |
|---|---|---|---|
| Prefer classical music ($C$) | 23% | 39% | 62% |
| Do not prefer classical music ($\sim C$) | 14% | 24% | 38% |
| Total | 37% | 63% | 100% |
Step6: Calculate the conditional probability for part b
The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Here, $A$ is the event of preferring classical music and $B$ is the event of being not left - handed. $P(A\cap B) = 39%$ and $P(B)=63%$. So $P(A|B)=\frac{39}{63}=\frac{13}{21}\approx 0.619$ or 61.9%
Answer:
a.
| Left - handed ($L$) | Not left - handed ($\sim L$) | Total | |
|---|---|---|---|
| Prefer classical music ($C$) | 23% | 39% | 62% |
| Do not prefer classical music ($\sim C$) | 14% | 24% | 38% |
| Total | 37% | 63% | 100% |
| b. $\frac{13}{21}\approx 61.9%$ |