To analyze the relative frequencies of music genre preferences among the 10th and 11th graders based on the given table, we can compute the relative frequency of each preference by each grade.
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Total Students:
- 10th graders: 165
- 11th graders: 245
- Total: 410
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Hip-Hop Preferences:
- 10th graders who prefer hip-hop: 112
- 11th graders who prefer hip-hop: 98
- Total hip-hop preferrers: 210
Relative frequency for hip-hop by grade:
- For 10th graders: \( \frac{112}{165} \approx 0.6788 \)
- For 11th graders: \( \frac{98}{245} \approx 0.4 \)
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Pop Preferences:
- 10th graders who prefer pop: 53
- 11th graders who prefer pop: 147
- Total pop preferrers: 200
Relative frequency for pop by grade:
- For 10th graders: \( \frac{53}{165} \approx 0.3212 \)
- For 11th graders: \( \frac{147}{245} \approx 0.6 \)
Now we can evaluate each statement based on the relative frequencies calculated:
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"If you like hip-hop, you are almost equally likely to be from 10th or 11th grade."
- Hip-hop preference distribution: 112 (10th)/210 ≈ 0.5333 from 10th grade and 98 (11th)/210 ≈ 0.4667 from 11th grade. This indicates a significant difference, so this statement is false.
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"If you are a 10th grader, you are more likely to prefer pop."
- 10th graders preferring pop: 53/165 ≈ 0.3212. The majority of 10th graders (112 out of 165) prefer hip-hop, thus this statement is false.
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"If you like pop, you are more likely to be a 10th grader."
- Pop preference distribution: 53 (10th)/200 ≈ 0.265 from 10th grade and 147 (11th)/200 ≈ 0.735 from 11th grade. This indicates that pop preference is stronger among 11th graders, so this statement is false.
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"If you are an 11th grader, you are more likely to prefer hip-hop."
- 11th graders who prefer hip-hop is 98 out of 245, and since 147 11th graders prefer pop, hip-hop is less preferred in this group. Thus this statement is false.
In conclusion, all statements provided are false based on the relative frequencies calculated from the data.