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Sagot :
To determine the coefficient of skewness and its interpretation for the given data set, we will follow a step-by-step approach:
### Step 1: List the given data and frequencies
The data values [tex]\( X \)[/tex] and their corresponding frequencies are:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline X & 3 & 4 & 5 & 6 & 7 \\ \hline \text{Frequency} & 3 & 2 & 4 & 2 & 3 \\ \hline \end{array} \][/tex]
### Step 2: Calculate the mean
The mean [tex]\(\mu\)[/tex] can be calculated using the formula:
[tex]\[ \mu = \frac{\sum (f \cdot X)}{\sum f} \][/tex]
Calculate [tex]\(\sum (f \cdot X)\)[/tex]:
[tex]\[ 3 \cdot 3 + 2 \cdot 4 + 4 \cdot 5 + 2 \cdot 6 + 3 \cdot 7 = 9 + 8 + 20 + 12 + 21 = 70 \][/tex]
Calculate [tex]\(\sum f\)[/tex]:
[tex]\[ 3 + 2 + 4 + 2 + 3 = 14 \][/tex]
Now, calculate the mean:
[tex]\[ \mu = \frac{70}{14} = 5 \][/tex]
### Step 3: Calculate the standard deviation
The standard deviation [tex]\(\sigma\)[/tex] is calculated using the formula:
[tex]\[ \sigma = \sqrt{\frac{\sum f (X - \mu)^2}{\sum f}} \][/tex]
Calculate [tex]\(\sum f (X - \mu)^2\)[/tex]:
[tex]\[ 3 (3 - 5)^2 + 2 (4 - 5)^2 + 4 (5 - 5)^2 + 2 (6 - 5)^2 + 3 (7 - 5)^2 \][/tex]
[tex]\[ 3 (2)^2 + 2 (1)^2 + 4 (0)^2 + 2 (1)^2 + 3 (2)^2 \][/tex]
[tex]\[ 3 \cdot 4 + 2 \cdot 1 + 4 \cdot 0 + 2 \cdot 1 + 3 \cdot 4 \][/tex]
[tex]\[ 12 + 2 + 0 + 2 + 12 = 28 \][/tex]
Now, calculate the variance:
[tex]\[ \text{Variance} = \frac{28}{14} = 2 \][/tex]
Hence, the standard deviation is:
[tex]\[ \sigma = \sqrt{2} \approx 1.414 \][/tex]
### Step 4: Calculate the skewness
The coefficient of skewness is calculated using the formula:
[tex]\[ \text{Skewness} = \frac{\sum f (X - \mu)^3}{(n \cdot \sigma^3)} \][/tex]
Where [tex]\( n \)[/tex] is the total frequency. Calculate [tex]\(\sigma^3\)[/tex]:
[tex]\[ \sigma^3 = 1.414^3 \approx 2.827 \][/tex]
Calculate [tex]\(\sum f (X - \mu)^3\)[/tex]:
[tex]\[ 3 (3 - 5)^3 + 2 (4 - 5)^3 + 4 (5 - 5)^3 + 2 (6 - 5)^3 + 3 (7 - 5)^3 \][/tex]
[tex]\[ 3 (-2)^3 + 2 (-1)^3 + 4 (0)^3 + 2 (1)^3 + 3 (2)^3 \][/tex]
[tex]\[ 3 (-8) + 2 (-1) + 4 (0) + 2 (1) + 3 (8) \][/tex]
[tex]\[ -24 - 2 + 0 + 2 + 24 = 0 \][/tex]
Now, calculate the skewness:
[tex]\[ \text{Skewness} = \frac{0}{14 \cdot 2.827} = 0 \][/tex]
### Step 5: Interpret the skewness
Since the skewness is 0, it indicates that the distribution is symmetrical.
### Conclusion
The coefficient of skewness is [tex]\( 0 \)[/tex], and its interpretation is "0 symmetrical distribution."
Therefore, the correct answer is:
A. 0 symmetrical distribution
### Step 1: List the given data and frequencies
The data values [tex]\( X \)[/tex] and their corresponding frequencies are:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline X & 3 & 4 & 5 & 6 & 7 \\ \hline \text{Frequency} & 3 & 2 & 4 & 2 & 3 \\ \hline \end{array} \][/tex]
### Step 2: Calculate the mean
The mean [tex]\(\mu\)[/tex] can be calculated using the formula:
[tex]\[ \mu = \frac{\sum (f \cdot X)}{\sum f} \][/tex]
Calculate [tex]\(\sum (f \cdot X)\)[/tex]:
[tex]\[ 3 \cdot 3 + 2 \cdot 4 + 4 \cdot 5 + 2 \cdot 6 + 3 \cdot 7 = 9 + 8 + 20 + 12 + 21 = 70 \][/tex]
Calculate [tex]\(\sum f\)[/tex]:
[tex]\[ 3 + 2 + 4 + 2 + 3 = 14 \][/tex]
Now, calculate the mean:
[tex]\[ \mu = \frac{70}{14} = 5 \][/tex]
### Step 3: Calculate the standard deviation
The standard deviation [tex]\(\sigma\)[/tex] is calculated using the formula:
[tex]\[ \sigma = \sqrt{\frac{\sum f (X - \mu)^2}{\sum f}} \][/tex]
Calculate [tex]\(\sum f (X - \mu)^2\)[/tex]:
[tex]\[ 3 (3 - 5)^2 + 2 (4 - 5)^2 + 4 (5 - 5)^2 + 2 (6 - 5)^2 + 3 (7 - 5)^2 \][/tex]
[tex]\[ 3 (2)^2 + 2 (1)^2 + 4 (0)^2 + 2 (1)^2 + 3 (2)^2 \][/tex]
[tex]\[ 3 \cdot 4 + 2 \cdot 1 + 4 \cdot 0 + 2 \cdot 1 + 3 \cdot 4 \][/tex]
[tex]\[ 12 + 2 + 0 + 2 + 12 = 28 \][/tex]
Now, calculate the variance:
[tex]\[ \text{Variance} = \frac{28}{14} = 2 \][/tex]
Hence, the standard deviation is:
[tex]\[ \sigma = \sqrt{2} \approx 1.414 \][/tex]
### Step 4: Calculate the skewness
The coefficient of skewness is calculated using the formula:
[tex]\[ \text{Skewness} = \frac{\sum f (X - \mu)^3}{(n \cdot \sigma^3)} \][/tex]
Where [tex]\( n \)[/tex] is the total frequency. Calculate [tex]\(\sigma^3\)[/tex]:
[tex]\[ \sigma^3 = 1.414^3 \approx 2.827 \][/tex]
Calculate [tex]\(\sum f (X - \mu)^3\)[/tex]:
[tex]\[ 3 (3 - 5)^3 + 2 (4 - 5)^3 + 4 (5 - 5)^3 + 2 (6 - 5)^3 + 3 (7 - 5)^3 \][/tex]
[tex]\[ 3 (-2)^3 + 2 (-1)^3 + 4 (0)^3 + 2 (1)^3 + 3 (2)^3 \][/tex]
[tex]\[ 3 (-8) + 2 (-1) + 4 (0) + 2 (1) + 3 (8) \][/tex]
[tex]\[ -24 - 2 + 0 + 2 + 24 = 0 \][/tex]
Now, calculate the skewness:
[tex]\[ \text{Skewness} = \frac{0}{14 \cdot 2.827} = 0 \][/tex]
### Step 5: Interpret the skewness
Since the skewness is 0, it indicates that the distribution is symmetrical.
### Conclusion
The coefficient of skewness is [tex]\( 0 \)[/tex], and its interpretation is "0 symmetrical distribution."
Therefore, the correct answer is:
A. 0 symmetrical distribution
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