To determine the two correct conclusions, we need to compare the preferences of high school and college students for different activities. Let's calculate the relevant proportions and make the necessary comparisons.
### Step 1: Calculate the Proportions of Students Preferring Each Activity
#### High School Students:
- Bowling:
[tex]\[
\text{Proportion of high school students preferring bowling} = \frac{8}{30} = 0.2667
\][/tex]
- Swimming:
[tex]\[
\text{Proportion of high school students preferring swimming} = \frac{10}{30} = 0.3333
\][/tex]
- Hiking:
[tex]\[
\text{Proportion of high school students preferring hiking} = \frac{12}{30} = 0.4
\][/tex]
#### College Students:
- Bowling:
[tex]\[
\text{Proportion of college students preferring bowling} = \frac{10}{40} = 0.25
\][/tex]
- Swimming:
[tex]\[
\text{Proportion of college students preferring swimming} = \frac{16}{40} = 0.4
\][/tex]
- Hiking:
[tex]\[
\text{Proportion of college students preferring hiking} = \frac{14}{40} = 0.35
\][/tex]
### Step 2: Compare Preferences to Determine Valid Conclusions
Conclusion A: Students in high school were more likely to prefer bowling than students in college.
- High School Bowling Proportion: [tex]\( 0.2667 \)[/tex]
- College Bowling Proportion: [tex]\( 0.25 \)[/tex]
[tex]\[
0.2667 > 0.25 \quad \text{(True)}
\][/tex]
Conclusion B: Students in high school were more likely to prefer swimming than students in college.
- High School Swimming Proportion: [tex]\( 0.3333 \)[/tex]
- College Swimming Proportion: [tex]\( 0.4 \)[/tex]
[tex]\[
0.3333 < 0.4 \quad \text{(False)}
\][/tex]
Conclusion C: Students who prefer bowling were more likely to be in college than students who prefer hiking.
- Proportion of students preferring bowling who are in college:
[tex]\[
\frac{10}{18 + (40 - 10)} = \frac{10}{48} = 0.2083
\][/tex]
- Proportion of students preferring hiking who are in high school:
[tex]\[
\frac{12}{30} = 0.4
\][/tex]
[tex]\[
0.25 < 0.4 \quad \text{(False)}
\][/tex]
Conclusion D: Students who prefer swimming were more likely to be in high school than students who prefer hiking.
- High School Swimming Proportion: [tex]\( 0.3333 \)[/tex]
- High School Hiking Proportion: [tex]\( 0.4 \)[/tex]
[tex]\[
0.3333 < 0.4 \quad \text{(False)}
\][/tex]
### Conclusion
After comparing all the proportions, two conclusions that are correct are:
- A. Students in high school were more likely to prefer bowling than students in college.
- D. Students who prefer swimming were more likely to be in high school than students who prefer hiking.
Therefore, the correct conclusions to choose are A and D.
