Answer :
To solve this problem, we will address each part step-by-step.
### Part A: Calculate the Measures of Center
Mean:
The mean (average) is calculated by summing all the values and dividing by the number of values.
For Mountain View School:
Sum of values \(= 9 + 8 + 2 + 0 + 8 + 7 + 6 + 5 + 5 + 4 + 4 + 3 + 1 + 0 = 62\)
Number of values \(= 14\)
[tex]\[ \text{Mean}_{\text{MV}} = \frac{62}{14} = 4.43 \][/tex]
For Bay Side School:
Sum of values \(= 5 + 6 + 8 + 0 + 2 + 4 + 5 + 6 + 8 + 0 + 0 + 2 + 3 + 5 + 2 = 56\)
Number of values \(= 15\)
[tex]\[ \text{Mean}_{\text{BS}} = \frac{56}{15} = 3.73 \][/tex]
Median:
The median is the middle number in a sorted list of numbers. If the number of observations is even, the median is the average of the two middle values.
For Mountain View School (sorted): \(0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 6, 7, 8, 9\)
Since there are 14 values, the median is the average of the 7th and 8th values.
[tex]\[ \text{Median}_{\text{MV}} = \frac{4 + 4}{2} = 4.5 \][/tex]
For Bay Side School (sorted): \(0, 0, 0, 2, 2, 3, 4, 5, 5, 6, 6, 8, 8, 5, 2\)
There are 15 values, so the median is the 8th value.
[tex]\[ \text{Median}_{\text{BS}} = 4.0 \][/tex]
### Part B: Calculate the Measures of Variability
Standard Deviation:
The standard deviation measures how spread out the numbers are from the mean.
For Mountain View School:
The values are \(9, 8, 2, 0, 8, 7, 6, 5, 5, 4, 4, 3, 1, 0\), and the mean is \(4.43\).
The standard deviation is calculated by the formula:
[tex]\[ s = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2} \][/tex]
Where \(N\) is the number of values, \(x_i\) are the values, and \(\overline{x}\) is the mean.
[tex]\[ s_{\text{MV}} = 2.98 \][/tex]
For Bay Side School:
The values are \(5, 6, 8, 0, 2, 4, 5, 6, 8, 0, 0, 2, 3, 5, 2\), and the mean is \(3.73\).
[tex]\[ s_{\text{BS}} = 2.71 \][/tex]
### Part C: Determining the Better School for Larger Class Size
To determine which school is better for larger class size, we compare their means. The school with the higher mean has a tendency to have larger class sizes.
From our calculations:
- Mean of class sizes at Mountain View School: \(4.43\)
- Mean of class sizes at Bay Side School: \(3.73\)
Since Mountain View School has a higher mean class size than Bay Side School, Mountain View School is the better choice if you are interested in larger class sizes.
Conclusion:
Mountain View School is a better choice for larger class sizes, as it has a higher mean score (4.43) compared to Bay Side School's mean score (3.73).
### Part A: Calculate the Measures of Center
Mean:
The mean (average) is calculated by summing all the values and dividing by the number of values.
For Mountain View School:
Sum of values \(= 9 + 8 + 2 + 0 + 8 + 7 + 6 + 5 + 5 + 4 + 4 + 3 + 1 + 0 = 62\)
Number of values \(= 14\)
[tex]\[ \text{Mean}_{\text{MV}} = \frac{62}{14} = 4.43 \][/tex]
For Bay Side School:
Sum of values \(= 5 + 6 + 8 + 0 + 2 + 4 + 5 + 6 + 8 + 0 + 0 + 2 + 3 + 5 + 2 = 56\)
Number of values \(= 15\)
[tex]\[ \text{Mean}_{\text{BS}} = \frac{56}{15} = 3.73 \][/tex]
Median:
The median is the middle number in a sorted list of numbers. If the number of observations is even, the median is the average of the two middle values.
For Mountain View School (sorted): \(0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 6, 7, 8, 9\)
Since there are 14 values, the median is the average of the 7th and 8th values.
[tex]\[ \text{Median}_{\text{MV}} = \frac{4 + 4}{2} = 4.5 \][/tex]
For Bay Side School (sorted): \(0, 0, 0, 2, 2, 3, 4, 5, 5, 6, 6, 8, 8, 5, 2\)
There are 15 values, so the median is the 8th value.
[tex]\[ \text{Median}_{\text{BS}} = 4.0 \][/tex]
### Part B: Calculate the Measures of Variability
Standard Deviation:
The standard deviation measures how spread out the numbers are from the mean.
For Mountain View School:
The values are \(9, 8, 2, 0, 8, 7, 6, 5, 5, 4, 4, 3, 1, 0\), and the mean is \(4.43\).
The standard deviation is calculated by the formula:
[tex]\[ s = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2} \][/tex]
Where \(N\) is the number of values, \(x_i\) are the values, and \(\overline{x}\) is the mean.
[tex]\[ s_{\text{MV}} = 2.98 \][/tex]
For Bay Side School:
The values are \(5, 6, 8, 0, 2, 4, 5, 6, 8, 0, 0, 2, 3, 5, 2\), and the mean is \(3.73\).
[tex]\[ s_{\text{BS}} = 2.71 \][/tex]
### Part C: Determining the Better School for Larger Class Size
To determine which school is better for larger class size, we compare their means. The school with the higher mean has a tendency to have larger class sizes.
From our calculations:
- Mean of class sizes at Mountain View School: \(4.43\)
- Mean of class sizes at Bay Side School: \(3.73\)
Since Mountain View School has a higher mean class size than Bay Side School, Mountain View School is the better choice if you are interested in larger class sizes.
Conclusion:
Mountain View School is a better choice for larger class sizes, as it has a higher mean score (4.43) compared to Bay Side School's mean score (3.73).
