To estimate the percentage of students that are already recycling based on the sample surveys provided, we will first calculate the recycling rates for each of the three halls surveyed, and then take the average of those rates.
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First Hall:
- Students Recycling: 32
- Total Students: 80
- Recycling Rate = \( \frac{32}{80} = 0.4 \) or 40%
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Second Hall:
- Students Recycling: 48
- Total Students: 110
- Recycling Rate = \( \frac{48}{110} \approx 0.4364 \) or 43.64%
-
Third Hall:
- Students Recycling: 49
- Total Students: 85
- Recycling Rate = \( \frac{49}{85} \approx 0.5765 \) or 57.65%
Now, we can convert these rates into percentages:
- Hall 1: 40%
- Hall 2: 43.64% (approximately 44%)
- Hall 3: 57.65% (approximately 58%)
Next, we calculate the weighted average based on the number of students in each hall (the total number of students in the sample):
- Total students surveyed: \( 80 + 110 + 85 = 275 \)
Now we calculate the contribution of each hall based on its recycling percentage:
- Contribution from Hall 1: \( 40% \times 80 = 32 \)
- Contribution from Hall 2: \( 43.64% \times 110 \approx 48 \)
- Contribution from Hall 3: \( 57.65% \times 85 \approx 49 \)
Now we sum these contributions to find the total number of students who are recycling in the sample:
- Total Recycling = \( 32 + 48 + 49 = 129 \)
Now we calculate the overall recycling rate:
\[ \text{Overall Recycling Rate} = \frac{\text{Total Recycling}}{\text{Total Students}} = \frac{129}{275} \approx 0.4673 \text{ or } 46.73% \]
Rounding this value gives approximately 47%.
Thus, the best estimate of the percentage of students that are already recycling is 47%.