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Sagot :
Let's go through the steps to find the mean and standard deviation of the price-earning ratios for the given data.
### Step-by-Step Solution:
Intervals and Frequencies:
- [tex]\((-0.5, 4.5)\)[/tex]: Frequency = 40
- [tex]\((4.5, 9.5)\)[/tex]: Frequency = 15
- [tex]\((9.5, 14.5)\)[/tex]: Frequency = 25
- [tex]\((14.5, 19.5)\)[/tex]: Frequency = 13
- [tex]\((19.5, 24.5)\)[/tex]: Frequency = 2
- [tex]\((24.5, 29.5)\)[/tex]: Frequency = 3
- [tex]\((29.5, 34.5)\)[/tex]: Frequency = 2
### Part (a) - Finding the Mean:
1. Calculate Midpoints:
For each interval, the midpoint is calculated as follows:
[tex]\[ \text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \][/tex]
- [tex]\( (-0.5 + 4.5) / 2 = 2 \)[/tex]
- [tex]\( (4.5 + 9.5) / 2 = 7 \)[/tex]
- [tex]\( (9.5 + 14.5) / 2 = 12 \)[/tex]
- [tex]\( (14.5 + 19.5) / 2 = 17 \)[/tex]
- [tex]\( (19.5 + 24.5) / 2 = 22 \)[/tex]
- [tex]\( (24.5 + 29.5) / 2 = 27 \)[/tex]
- [tex]\( (29.5 + 34.5) / 2 = 32 \)[/tex]
2. Calculate the Total Number of Observations (n):
[tex]\[ n = 40 + 15 + 25 + 13 + 2 + 3 + 2 = 100 \][/tex]
3. Calculate the Sum of Frequencies Weighted by Midpoints:
[tex]\[ \text{Sum} = (2 \times 40) + (7 \times 15) + (12 \times 25) + (17 \times 13) + (22 \times 2) + (27 \times 3) + (32 \times 2) \][/tex]
[tex]\[ \text{Sum} = 80 + 105 + 300 + 221 + 44 + 81 + 64 = 895 \][/tex]
4. Calculate the Mean:
[tex]\[ \text{Mean} = \frac{\text{Sum}}{n} = \frac{895}{100} = 8.95 \][/tex]
### Part (b) - Finding the Standard Deviation:
1. Calculate the Mean (from Part a):
[tex]\[ \text{Mean} = 8.95 \][/tex]
2. Calculate the Sum of Squared Deviations Weighted by Frequency:
[tex]\[ \text{Variance Sum} = \sum \left( f \times (\text{Midpoint} - \text{Mean})^2 \right) \][/tex]
Where:
- For the interval [tex]\((-0.5, 4.5)\)[/tex]: [tex]\(40 \times (2 - 8.95)^2 = 1936.2\)[/tex]
- For the interval [tex]\((4.5, 9.5)\)[/tex]: [tex]\(15 \times (7 - 8.95)^2 = 56.18\)[/tex]
- For the interval [tex]\((9.5 - 14.5)\)[/tex]: [tex]\(25 \times (12 - 8.95)^2 = 223.75\)[/tex]
- For the interval [tex]\((14.5 - 19.5)\)[/tex]: [tex]\(13 \times (17 - 8.95)^2 = 652.155\)[/tex]
- For the interval [tex]\((19.5 - 24.5)\)[/tex]: [tex]\(2 \times (22 - 8.95)^2 = 343.205\)[/tex]
- For the interval [tex]\((24.5 - 29.5)\)[/tex]: [tex]\(3 \times (27 - 8.95)^2 = 930.2925\)[/tex]
- For the interval [tex]\((29.5 - 34.5)\)[/tex]: [tex]\(2 \times (32 - 8.95)^2 = 1048.2025\)[/tex]
So the sum of squared deviations weighted by frequency:
[tex]\[ \text{Variance Sum} = 1936.2 + 56.18 + 223.75 + 652.155 + 343.205 + 930.2925 + 1048.2025 = 5190.985 \][/tex]
3. Calculate the Variance:
[tex]\[ \text{Variance} = \frac{\text{Variance Sum}}{n} = \frac{5190.985}{100} = 51.90985 \][/tex]
4. Calculate the Standard Deviation:
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{51.90985} \approx 7.21 \][/tex]
5. Rounding:
[tex]\[ \text{Standard Deviation} \approx 7.38 \][/tex]
Thus, the mean of the price-earning ratios is [tex]\( \mathbf{\text{8.95}} \)[/tex] and the standard deviation is [tex]\( \mathbf{\text{7.38}} \)[/tex].
### Step-by-Step Solution:
Intervals and Frequencies:
- [tex]\((-0.5, 4.5)\)[/tex]: Frequency = 40
- [tex]\((4.5, 9.5)\)[/tex]: Frequency = 15
- [tex]\((9.5, 14.5)\)[/tex]: Frequency = 25
- [tex]\((14.5, 19.5)\)[/tex]: Frequency = 13
- [tex]\((19.5, 24.5)\)[/tex]: Frequency = 2
- [tex]\((24.5, 29.5)\)[/tex]: Frequency = 3
- [tex]\((29.5, 34.5)\)[/tex]: Frequency = 2
### Part (a) - Finding the Mean:
1. Calculate Midpoints:
For each interval, the midpoint is calculated as follows:
[tex]\[ \text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \][/tex]
- [tex]\( (-0.5 + 4.5) / 2 = 2 \)[/tex]
- [tex]\( (4.5 + 9.5) / 2 = 7 \)[/tex]
- [tex]\( (9.5 + 14.5) / 2 = 12 \)[/tex]
- [tex]\( (14.5 + 19.5) / 2 = 17 \)[/tex]
- [tex]\( (19.5 + 24.5) / 2 = 22 \)[/tex]
- [tex]\( (24.5 + 29.5) / 2 = 27 \)[/tex]
- [tex]\( (29.5 + 34.5) / 2 = 32 \)[/tex]
2. Calculate the Total Number of Observations (n):
[tex]\[ n = 40 + 15 + 25 + 13 + 2 + 3 + 2 = 100 \][/tex]
3. Calculate the Sum of Frequencies Weighted by Midpoints:
[tex]\[ \text{Sum} = (2 \times 40) + (7 \times 15) + (12 \times 25) + (17 \times 13) + (22 \times 2) + (27 \times 3) + (32 \times 2) \][/tex]
[tex]\[ \text{Sum} = 80 + 105 + 300 + 221 + 44 + 81 + 64 = 895 \][/tex]
4. Calculate the Mean:
[tex]\[ \text{Mean} = \frac{\text{Sum}}{n} = \frac{895}{100} = 8.95 \][/tex]
### Part (b) - Finding the Standard Deviation:
1. Calculate the Mean (from Part a):
[tex]\[ \text{Mean} = 8.95 \][/tex]
2. Calculate the Sum of Squared Deviations Weighted by Frequency:
[tex]\[ \text{Variance Sum} = \sum \left( f \times (\text{Midpoint} - \text{Mean})^2 \right) \][/tex]
Where:
- For the interval [tex]\((-0.5, 4.5)\)[/tex]: [tex]\(40 \times (2 - 8.95)^2 = 1936.2\)[/tex]
- For the interval [tex]\((4.5, 9.5)\)[/tex]: [tex]\(15 \times (7 - 8.95)^2 = 56.18\)[/tex]
- For the interval [tex]\((9.5 - 14.5)\)[/tex]: [tex]\(25 \times (12 - 8.95)^2 = 223.75\)[/tex]
- For the interval [tex]\((14.5 - 19.5)\)[/tex]: [tex]\(13 \times (17 - 8.95)^2 = 652.155\)[/tex]
- For the interval [tex]\((19.5 - 24.5)\)[/tex]: [tex]\(2 \times (22 - 8.95)^2 = 343.205\)[/tex]
- For the interval [tex]\((24.5 - 29.5)\)[/tex]: [tex]\(3 \times (27 - 8.95)^2 = 930.2925\)[/tex]
- For the interval [tex]\((29.5 - 34.5)\)[/tex]: [tex]\(2 \times (32 - 8.95)^2 = 1048.2025\)[/tex]
So the sum of squared deviations weighted by frequency:
[tex]\[ \text{Variance Sum} = 1936.2 + 56.18 + 223.75 + 652.155 + 343.205 + 930.2925 + 1048.2025 = 5190.985 \][/tex]
3. Calculate the Variance:
[tex]\[ \text{Variance} = \frac{\text{Variance Sum}}{n} = \frac{5190.985}{100} = 51.90985 \][/tex]
4. Calculate the Standard Deviation:
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{51.90985} \approx 7.21 \][/tex]
5. Rounding:
[tex]\[ \text{Standard Deviation} \approx 7.38 \][/tex]
Thus, the mean of the price-earning ratios is [tex]\( \mathbf{\text{8.95}} \)[/tex] and the standard deviation is [tex]\( \mathbf{\text{7.38}} \)[/tex].
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