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Sagot :
Sure! Let's go through each part step by step.
### Part A: Calculate the measures of center
We need to calculate the mean and median for each school.
1. Siry View School:
- Data from the stem-and-leaf plot: [tex]\(0, 9, 7, 2, 0, 18, 17, 16, 15, 15, 15, 14, 13, 11, 10, 20\)[/tex]
- Mean:
[tex]\[ \text{Mean} = \frac{\sum \text{data}}{\text{number of data points}} = \frac{182}{16} = 11.375 \][/tex]
- Median:
- Sorting the data: [tex]\(0, 0, 2, 7, 9, 10, 11, 13, 14, 15, 15, 15, 16, 17, 18, 20\)[/tex]
- The median is the average of the 8th and 9th values:
[tex]\[ \text{Median} = \frac{13 + 14}{2} = 13.5 \][/tex]
2. South Lake School:
- Data from the stem-and-leaf plot: [tex]\(5, 8, 10, 11, 12, 16, 16, 13, 25, 25, 26, 27, 28, 30, 36\)[/tex]
- Mean:
[tex]\[ \text{Mean} = \frac{\sum \text{data}}{\text{number of data points}} = \frac{288}{15} = 19.2 \][/tex]
- Median:
- Sorting the data: [tex]\(5, 8, 10, 11, 12, 13, 16, 16, 25, 25, 26, 27, 28, 30, 36\)[/tex]
- The median is the 8th value:
[tex]\[ \text{Median} = 16.0 \][/tex]
### Part B: Calculate the measures of variability
We need to calculate the variance and standard deviation.
1. Siry View School:
- Data: [tex]\(0, 9, 7, 2, 0, 18, 17, 16, 15, 15, 15, 14, 13, 11, 10, 20\)[/tex]
- Variance ([tex]\(s^2\)[/tex]):
[tex]\[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = 39.583 \][/tex]
- Standard Deviation ([tex]\(s\)[/tex]):
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{39.583} \approx 6.292 \][/tex]
2. South Lake School:
- Data: [tex]\(5, 8, 10, 11, 12, 16, 16, 13, 25, 25, 26, 27, 28, 30, 36\)[/tex]
- Variance ([tex]\(s^2\)[/tex]):
[tex]\[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = 88.6 \][/tex]
- Standard Deviation ([tex]\(s\)[/tex]):
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{88.6} \approx 9.413 \][/tex]
### Part C: Determine the better school for smaller class size
To determine the better school based on smaller class sizes, we compare the means of the two schools.
- Siry View School has a mean class size of [tex]\(11.375\)[/tex]
- South Lake School has a mean class size of [tex]\(19.2\)[/tex]
Since [tex]\(11.375\)[/tex] (Siry View) is less than [tex]\(19.2\)[/tex] (South Lake), Siry View has smaller class sizes on average.
### Conclusion
Hence, if you are interested in a smaller class size, Siry View is the better choice. This conclusion is based on the lower mean class size observed at Siry View compared to South Lake.
### Part A: Calculate the measures of center
We need to calculate the mean and median for each school.
1. Siry View School:
- Data from the stem-and-leaf plot: [tex]\(0, 9, 7, 2, 0, 18, 17, 16, 15, 15, 15, 14, 13, 11, 10, 20\)[/tex]
- Mean:
[tex]\[ \text{Mean} = \frac{\sum \text{data}}{\text{number of data points}} = \frac{182}{16} = 11.375 \][/tex]
- Median:
- Sorting the data: [tex]\(0, 0, 2, 7, 9, 10, 11, 13, 14, 15, 15, 15, 16, 17, 18, 20\)[/tex]
- The median is the average of the 8th and 9th values:
[tex]\[ \text{Median} = \frac{13 + 14}{2} = 13.5 \][/tex]
2. South Lake School:
- Data from the stem-and-leaf plot: [tex]\(5, 8, 10, 11, 12, 16, 16, 13, 25, 25, 26, 27, 28, 30, 36\)[/tex]
- Mean:
[tex]\[ \text{Mean} = \frac{\sum \text{data}}{\text{number of data points}} = \frac{288}{15} = 19.2 \][/tex]
- Median:
- Sorting the data: [tex]\(5, 8, 10, 11, 12, 13, 16, 16, 25, 25, 26, 27, 28, 30, 36\)[/tex]
- The median is the 8th value:
[tex]\[ \text{Median} = 16.0 \][/tex]
### Part B: Calculate the measures of variability
We need to calculate the variance and standard deviation.
1. Siry View School:
- Data: [tex]\(0, 9, 7, 2, 0, 18, 17, 16, 15, 15, 15, 14, 13, 11, 10, 20\)[/tex]
- Variance ([tex]\(s^2\)[/tex]):
[tex]\[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = 39.583 \][/tex]
- Standard Deviation ([tex]\(s\)[/tex]):
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{39.583} \approx 6.292 \][/tex]
2. South Lake School:
- Data: [tex]\(5, 8, 10, 11, 12, 16, 16, 13, 25, 25, 26, 27, 28, 30, 36\)[/tex]
- Variance ([tex]\(s^2\)[/tex]):
[tex]\[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = 88.6 \][/tex]
- Standard Deviation ([tex]\(s\)[/tex]):
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{88.6} \approx 9.413 \][/tex]
### Part C: Determine the better school for smaller class size
To determine the better school based on smaller class sizes, we compare the means of the two schools.
- Siry View School has a mean class size of [tex]\(11.375\)[/tex]
- South Lake School has a mean class size of [tex]\(19.2\)[/tex]
Since [tex]\(11.375\)[/tex] (Siry View) is less than [tex]\(19.2\)[/tex] (South Lake), Siry View has smaller class sizes on average.
### Conclusion
Hence, if you are interested in a smaller class size, Siry View is the better choice. This conclusion is based on the lower mean class size observed at Siry View compared to South Lake.
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