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Sagot :
Sure, I'd be happy to help with these problems!
### Part (a): Calculate Mean, Median, Mode, and Mean Deviation
Given the marks distribution:
| MARKS | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|------------|----|----|----|----|----|----|----|----|
| FREQUENCY | 2 | 6 | 5 | 4 | 6 | 9 | 8 | 5 |
We first need to create the expanded data set based on the frequencies.
Expanded Data:
[tex]\[ [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9] \][/tex]
Step-by-Step Calculation:
1. Mean:
To find the mean, we sum up all the marks and then divide by the total number of observations.
[tex]\[ \text{Mean} = \frac{\text{Sum of all marks}}{\text{Total number of observations}} \][/tex]
Sum of all marks:
[tex]\[ 2\cdot2 + 3\cdot6 + 4\cdot5 + 5\cdot4 + 6\cdot6 + 7\cdot9 + 8\cdot8 + 9\cdot5 = 4 + 18 + 20 + 20 + 36 + 63 + 64 + 45 = 270 \][/tex]
Total number of observations:
[tex]\[ 2+6+5+4+6+9+8+5 = 45 \][/tex]
Mean:
[tex]\[ \text{Mean} = \frac{270}{45} = 6 \][/tex]
2. Median:
The median is the middle value of an ordered list. Since there are 45 observations (an odd number), the median is the value of the 23rd observation when all are ordered.
From the expanded data:
- There are 17 marks less than 6 (2 2 + 6 3 + 5 4 + 4 5).
- The next 6 marks are 6's.
The 23rd mark is thus a 6.
Median:
[tex]\[ \text{Median} = 6 \][/tex]
3. Mode:
The mode is the mark with the highest frequency.
From the frequency distribution:
- The mark 7 has the highest frequency of 9.
Mode:
[tex]\[ \text{Mode} = 7 \][/tex]
4. Mean Deviation:
To find the mean deviation, we first find the deviations of each mark from the mean, take their absolute values, and then find the average of these deviations.
Deviations:
[tex]\[ |2-6| = 4, \: |3-6| = 3, \: |4-6| = 2, \: |5-6| = 1, \: |6-6| = 0, \: |7-6| = 1, \: |8-6| = 2, \: |9-6| = 3 \][/tex]
Thus, the total deviation for each mark considering their frequencies will be:
[tex]\[ 2 \cdot 4 + 6 \cdot 3 + 5 \cdot 2 + 4 \cdot 1 + 6 \cdot 0 + 9 \cdot 1 + 8 \cdot 2 + 5 \cdot 3 = 8 + 18 + 10 + 4 + 0 + 9 + 16 + 15 = 80 \][/tex]
Mean Deviation:
[tex]\[ \text{Mean Deviation} = \frac{80}{45} \approx 1.778 \][/tex]
### Part (b): Construct a Quadratic Equation with Given Roots
Given roots are [tex]\(\frac{4}{5}\)[/tex] and [tex]\(-2 \frac{2}{3}\)[/tex].
Convert [tex]\(-2 \frac{2}{3}\)[/tex] into an improper fraction:
[tex]\[ -2 \frac{2}{3} = -\left(2 + \frac{2}{3}\right) = -\frac{6}{3} - \frac{2}{3} = -\frac{8}{3} \][/tex]
Let the roots of the quadratic equation be [tex]\( \alpha = \frac{4}{5} \)[/tex] and [tex]\(\beta = -\frac{8}{3}\)[/tex].
The quadratic equation with these roots can be expressed as:
[tex]\[ (x - \alpha)(x - \beta) = 0 \][/tex]
Expanding this, we use Vieta's formulas:
[tex]\[ x^2 - ( \alpha + \beta)x + \alpha \beta = 0 \][/tex]
Calculate [tex]\(\alpha + \beta\)[/tex]:
[tex]\[ \alpha + \beta = \frac{4}{5} + \left(-\frac{8}{3}\right) = \frac{4}{5} - \frac{8}{3} = \frac{4 \cdot 3 - 8 \cdot 5}{15} = \frac{12 - 40}{15} = -\frac{28}{15} \][/tex]
Calculate [tex]\(\alpha \beta\)[/tex]:
[tex]\[ \alpha \beta = \frac{4}{5} \cdot \left(-\frac{8}{3}\right) = -\frac{32}{15} \][/tex]
Thus, the quadratic equation is:
[tex]\[ x^2 - \left(-\frac{28}{15}\right)x + \left(-\frac{32}{15}\right) = 0 \][/tex]
Which simplifies to:
[tex]\[ x^2 + \frac{28}{15}x - \frac{32}{15} = 0 \][/tex]
Multiplying through by 15 to clear the fractions:
[tex]\[ 15x^2 + 28x - 32 = 0 \][/tex]
This is the quadratic equation whose roots are [tex]\(\frac{4}{5}\)[/tex] and [tex]\(-2 \frac{2}{3}\)[/tex].
### Summary
(a):
- Mean: 6
- Median: 6
- Mode: 7
- Mean Deviation: [tex]\(\approx 1.778\)[/tex]
(b):
- Quadratic Equation: [tex]\(15x^2 + 28x - 32 = 0\)[/tex]
### Part (a): Calculate Mean, Median, Mode, and Mean Deviation
Given the marks distribution:
| MARKS | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|------------|----|----|----|----|----|----|----|----|
| FREQUENCY | 2 | 6 | 5 | 4 | 6 | 9 | 8 | 5 |
We first need to create the expanded data set based on the frequencies.
Expanded Data:
[tex]\[ [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9] \][/tex]
Step-by-Step Calculation:
1. Mean:
To find the mean, we sum up all the marks and then divide by the total number of observations.
[tex]\[ \text{Mean} = \frac{\text{Sum of all marks}}{\text{Total number of observations}} \][/tex]
Sum of all marks:
[tex]\[ 2\cdot2 + 3\cdot6 + 4\cdot5 + 5\cdot4 + 6\cdot6 + 7\cdot9 + 8\cdot8 + 9\cdot5 = 4 + 18 + 20 + 20 + 36 + 63 + 64 + 45 = 270 \][/tex]
Total number of observations:
[tex]\[ 2+6+5+4+6+9+8+5 = 45 \][/tex]
Mean:
[tex]\[ \text{Mean} = \frac{270}{45} = 6 \][/tex]
2. Median:
The median is the middle value of an ordered list. Since there are 45 observations (an odd number), the median is the value of the 23rd observation when all are ordered.
From the expanded data:
- There are 17 marks less than 6 (2 2 + 6 3 + 5 4 + 4 5).
- The next 6 marks are 6's.
The 23rd mark is thus a 6.
Median:
[tex]\[ \text{Median} = 6 \][/tex]
3. Mode:
The mode is the mark with the highest frequency.
From the frequency distribution:
- The mark 7 has the highest frequency of 9.
Mode:
[tex]\[ \text{Mode} = 7 \][/tex]
4. Mean Deviation:
To find the mean deviation, we first find the deviations of each mark from the mean, take their absolute values, and then find the average of these deviations.
Deviations:
[tex]\[ |2-6| = 4, \: |3-6| = 3, \: |4-6| = 2, \: |5-6| = 1, \: |6-6| = 0, \: |7-6| = 1, \: |8-6| = 2, \: |9-6| = 3 \][/tex]
Thus, the total deviation for each mark considering their frequencies will be:
[tex]\[ 2 \cdot 4 + 6 \cdot 3 + 5 \cdot 2 + 4 \cdot 1 + 6 \cdot 0 + 9 \cdot 1 + 8 \cdot 2 + 5 \cdot 3 = 8 + 18 + 10 + 4 + 0 + 9 + 16 + 15 = 80 \][/tex]
Mean Deviation:
[tex]\[ \text{Mean Deviation} = \frac{80}{45} \approx 1.778 \][/tex]
### Part (b): Construct a Quadratic Equation with Given Roots
Given roots are [tex]\(\frac{4}{5}\)[/tex] and [tex]\(-2 \frac{2}{3}\)[/tex].
Convert [tex]\(-2 \frac{2}{3}\)[/tex] into an improper fraction:
[tex]\[ -2 \frac{2}{3} = -\left(2 + \frac{2}{3}\right) = -\frac{6}{3} - \frac{2}{3} = -\frac{8}{3} \][/tex]
Let the roots of the quadratic equation be [tex]\( \alpha = \frac{4}{5} \)[/tex] and [tex]\(\beta = -\frac{8}{3}\)[/tex].
The quadratic equation with these roots can be expressed as:
[tex]\[ (x - \alpha)(x - \beta) = 0 \][/tex]
Expanding this, we use Vieta's formulas:
[tex]\[ x^2 - ( \alpha + \beta)x + \alpha \beta = 0 \][/tex]
Calculate [tex]\(\alpha + \beta\)[/tex]:
[tex]\[ \alpha + \beta = \frac{4}{5} + \left(-\frac{8}{3}\right) = \frac{4}{5} - \frac{8}{3} = \frac{4 \cdot 3 - 8 \cdot 5}{15} = \frac{12 - 40}{15} = -\frac{28}{15} \][/tex]
Calculate [tex]\(\alpha \beta\)[/tex]:
[tex]\[ \alpha \beta = \frac{4}{5} \cdot \left(-\frac{8}{3}\right) = -\frac{32}{15} \][/tex]
Thus, the quadratic equation is:
[tex]\[ x^2 - \left(-\frac{28}{15}\right)x + \left(-\frac{32}{15}\right) = 0 \][/tex]
Which simplifies to:
[tex]\[ x^2 + \frac{28}{15}x - \frac{32}{15} = 0 \][/tex]
Multiplying through by 15 to clear the fractions:
[tex]\[ 15x^2 + 28x - 32 = 0 \][/tex]
This is the quadratic equation whose roots are [tex]\(\frac{4}{5}\)[/tex] and [tex]\(-2 \frac{2}{3}\)[/tex].
### Summary
(a):
- Mean: 6
- Median: 6
- Mode: 7
- Mean Deviation: [tex]\(\approx 1.778\)[/tex]
(b):
- Quadratic Equation: [tex]\(15x^2 + 28x - 32 = 0\)[/tex]
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