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Sagot :
Let's interpret and analyze the data provided in the table and address each of the given statements using relative frequencies and probabilities.
Table Summary:
- 10th Grade:
- Hip-Hop: 112
- Pop: 53
- Total Students: 165
- 11th Grade:
- Hip-Hop: 98
- Pop: 147
- Total Students: 245
- Overall:
- Hip-Hop Total: 210
- Pop Total: 200
- Total Students: 410
### Step-by-Step Analysis:
1. Probability if you are an 11th grader and prefer hip-hop:
- Number of 11th graders who like Hip-Hop: 98
- Total number of 11th graders: 245
- Relative frequency [tex]\( P(\text{Hip-Hop} | \text{11th Grade}) \)[/tex]:
[tex]\[ \frac{98}{245} = 0.4 = 40\% \][/tex]
- Therefore, the probability is 40%.
2. Probability if you like pop, you are more likely to be a 10th grader:
- Number of 10th graders who like Pop: 53
- Total number of students who like Pop: 200
- Relative frequency [tex]\( P(\text{10th Grade} | \text{Pop}) \)[/tex]:
[tex]\[ \frac{53}{200} = 0.265 = 26.5\% \][/tex]
- Therefore, the probability is 26.5%.
3. Probability if you are a 10th grader, you are more likely to prefer pop:
- Number of 10th graders who like Pop: 53
- Total number of 10th graders: 165
- Relative frequency [tex]\( P(\text{Pop} | \text{10th Grade}) \)[/tex]:
[tex]\[ \frac{53}{165} = 0.3212 \approx 32.1\% \][/tex]
- Therefore, the probability is approximately 32.1%.
4. Probability if you like hip-hop, you are almost equally likely to be from 10th or 11th grade:
- Number of 10th graders who like Hip-Hop: 112
- Number of 11th graders who like Hip-Hop: 98
- Total number of students who like Hip-Hop: 210
- Relative frequency [tex]\( P(\text{10th Grade} | \text{Hip-Hop}) \)[/tex]:
[tex]\[ \frac{112}{210} = 0.5333 \approx 53.3\% \][/tex]
- Relative frequency [tex]\( P(\text{11th Grade} | \text{Hip-Hop}) \)[/tex]:
[tex]\[ \frac{98}{210} = 0.4667 \approx 46.7\% \][/tex]
- These probabilities are quite close (53.3% vs. 46.7%), indicating that you are almost equally likely to be from either the 10th or 11th grade if you like Hip-Hop.
### Conclusion:
Based on the relative frequencies calculated:
- The first statement "If you are an 11th grader, you are more likely to prefer hip-hop" is true with a probability of 40%.
- The second statement "If you like pop, you are more likely to be a 10th grader" is false with a probability of only 26.5%.
- The third statement "If you are a 10th grader, you are more likely to prefer pop" is false with a probability of 32.1%, which is lower than 50%.
- The fourth statement "If you like hip-hop, you are almost equally likely to be from 10th or 11th grade" is true, with probabilities 53.3% and 46.7%, respectively.
Thus, the true statements based on the data and relative frequency analysis are:
- "If you are an 11th grader, you are more likely to prefer hip-hop".
- "If you like hip-hop, you are almost equally likely to be from 10th or 11th grade".
Table Summary:
- 10th Grade:
- Hip-Hop: 112
- Pop: 53
- Total Students: 165
- 11th Grade:
- Hip-Hop: 98
- Pop: 147
- Total Students: 245
- Overall:
- Hip-Hop Total: 210
- Pop Total: 200
- Total Students: 410
### Step-by-Step Analysis:
1. Probability if you are an 11th grader and prefer hip-hop:
- Number of 11th graders who like Hip-Hop: 98
- Total number of 11th graders: 245
- Relative frequency [tex]\( P(\text{Hip-Hop} | \text{11th Grade}) \)[/tex]:
[tex]\[ \frac{98}{245} = 0.4 = 40\% \][/tex]
- Therefore, the probability is 40%.
2. Probability if you like pop, you are more likely to be a 10th grader:
- Number of 10th graders who like Pop: 53
- Total number of students who like Pop: 200
- Relative frequency [tex]\( P(\text{10th Grade} | \text{Pop}) \)[/tex]:
[tex]\[ \frac{53}{200} = 0.265 = 26.5\% \][/tex]
- Therefore, the probability is 26.5%.
3. Probability if you are a 10th grader, you are more likely to prefer pop:
- Number of 10th graders who like Pop: 53
- Total number of 10th graders: 165
- Relative frequency [tex]\( P(\text{Pop} | \text{10th Grade}) \)[/tex]:
[tex]\[ \frac{53}{165} = 0.3212 \approx 32.1\% \][/tex]
- Therefore, the probability is approximately 32.1%.
4. Probability if you like hip-hop, you are almost equally likely to be from 10th or 11th grade:
- Number of 10th graders who like Hip-Hop: 112
- Number of 11th graders who like Hip-Hop: 98
- Total number of students who like Hip-Hop: 210
- Relative frequency [tex]\( P(\text{10th Grade} | \text{Hip-Hop}) \)[/tex]:
[tex]\[ \frac{112}{210} = 0.5333 \approx 53.3\% \][/tex]
- Relative frequency [tex]\( P(\text{11th Grade} | \text{Hip-Hop}) \)[/tex]:
[tex]\[ \frac{98}{210} = 0.4667 \approx 46.7\% \][/tex]
- These probabilities are quite close (53.3% vs. 46.7%), indicating that you are almost equally likely to be from either the 10th or 11th grade if you like Hip-Hop.
### Conclusion:
Based on the relative frequencies calculated:
- The first statement "If you are an 11th grader, you are more likely to prefer hip-hop" is true with a probability of 40%.
- The second statement "If you like pop, you are more likely to be a 10th grader" is false with a probability of only 26.5%.
- The third statement "If you are a 10th grader, you are more likely to prefer pop" is false with a probability of 32.1%, which is lower than 50%.
- The fourth statement "If you like hip-hop, you are almost equally likely to be from 10th or 11th grade" is true, with probabilities 53.3% and 46.7%, respectively.
Thus, the true statements based on the data and relative frequency analysis are:
- "If you are an 11th grader, you are more likely to prefer hip-hop".
- "If you like hip-hop, you are almost equally likely to be from 10th or 11th grade".
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