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To compute the sample variance and standard deviation of the given data sample, we will go through the following steps:
### Data Sample:
[tex]\[ 2, -3.8, 2.3, -0.7, -0.7 \][/tex]
### Step 1: Calculate the Mean
First, we calculate the mean (average) of the data points.
[tex]\[ \text{Mean} = \frac{2 + (-3.8) + 2.3 + (-0.7) + (-0.7)}{5} \][/tex]
[tex]\[ \text{Mean} = \frac{2 - 3.8 + 2.3 - 0.7 - 0.7}{5} \][/tex]
[tex]\[ \text{Mean} = \frac{-1.9}{5} \][/tex]
[tex]\[ \text{Mean} = -0.38 \][/tex]
### Step 2: Calculate Deviations from the Mean
Next, we find the deviation of each data point from the mean, square these deviations, and then sum them up.
[tex]\[ \begin{aligned} & \text{Deviation for } 2: (2 - (-0.38))^2 = (2 + 0.38)^2 = 2.38^2 \\ & \text{Deviation for } -3.8: (-3.8 - (-0.38))^2 = (-3.8 + 0.38)^2 = (-3.42)^2 \\ & \text{Deviation for } 2.3: (2.3 - (-0.38))^2 = (2.3 + 0.38)^2 = 2.68^2 \\ & \text{Deviation for } -0.7: (-0.7 - (-0.38))^2 = (-0.7 + 0.38)^2 = (-0.32)^2 \\ & \text{Deviation for } -0.7: (-0.7 - (-0.38))^2 = (-0.7 + 0.38)^2 = (-0.32)^2 \\ \end{aligned} \][/tex]
Perform the calculations:
[tex]\[ \begin{aligned} & 2.38^2 = 5.6644 \\ & (-3.42)^2 = 11.6964 \\ & 2.68^2 = 7.1824 \\ & (-0.32)^2 = 0.1024 \\ & (-0.32)^2 = 0.1024 \\ \end{aligned} \][/tex]
Now sum these squared deviations:
[tex]\[ 5.6644 + 11.6964 + 7.1824 + 0.1024 + 0.1024 = 24.748 \][/tex]
### Step 3: Compute the Sample Variance
The sample variance is calculated by dividing this sum by the number of data points minus one. Here, our sample size \( n = 5 \).
[tex]\[ \text{Sample Variance} = \frac{24.748}{5 - 1} \][/tex]
[tex]\[ \text{Sample Variance} = \frac{24.748}{4} \][/tex]
[tex]\[ \text{Sample Variance} = 6.187 \][/tex]
### Step 4: Compute the Standard Deviation
The standard deviation is the square root of the sample variance.
[tex]\[ \text{Standard Deviation} = \sqrt{6.187} \][/tex]
[tex]\[ \text{Standard Deviation} \approx 2.487 \][/tex]
### Step 5: Round to Two Decimal Places
Finally, we round the sample variance and standard deviation to two decimal places.
[tex]\[ \text{Sample Variance} \approx 6.14 \][/tex]
[tex]\[ \text{Standard Deviation} \approx 2.48 \][/tex]
### Results
Variance \( \square \) = \( 6.14 \)
Standard Deviation [tex]\( \square \)[/tex] = [tex]\( 2.48 \)[/tex]
### Data Sample:
[tex]\[ 2, -3.8, 2.3, -0.7, -0.7 \][/tex]
### Step 1: Calculate the Mean
First, we calculate the mean (average) of the data points.
[tex]\[ \text{Mean} = \frac{2 + (-3.8) + 2.3 + (-0.7) + (-0.7)}{5} \][/tex]
[tex]\[ \text{Mean} = \frac{2 - 3.8 + 2.3 - 0.7 - 0.7}{5} \][/tex]
[tex]\[ \text{Mean} = \frac{-1.9}{5} \][/tex]
[tex]\[ \text{Mean} = -0.38 \][/tex]
### Step 2: Calculate Deviations from the Mean
Next, we find the deviation of each data point from the mean, square these deviations, and then sum them up.
[tex]\[ \begin{aligned} & \text{Deviation for } 2: (2 - (-0.38))^2 = (2 + 0.38)^2 = 2.38^2 \\ & \text{Deviation for } -3.8: (-3.8 - (-0.38))^2 = (-3.8 + 0.38)^2 = (-3.42)^2 \\ & \text{Deviation for } 2.3: (2.3 - (-0.38))^2 = (2.3 + 0.38)^2 = 2.68^2 \\ & \text{Deviation for } -0.7: (-0.7 - (-0.38))^2 = (-0.7 + 0.38)^2 = (-0.32)^2 \\ & \text{Deviation for } -0.7: (-0.7 - (-0.38))^2 = (-0.7 + 0.38)^2 = (-0.32)^2 \\ \end{aligned} \][/tex]
Perform the calculations:
[tex]\[ \begin{aligned} & 2.38^2 = 5.6644 \\ & (-3.42)^2 = 11.6964 \\ & 2.68^2 = 7.1824 \\ & (-0.32)^2 = 0.1024 \\ & (-0.32)^2 = 0.1024 \\ \end{aligned} \][/tex]
Now sum these squared deviations:
[tex]\[ 5.6644 + 11.6964 + 7.1824 + 0.1024 + 0.1024 = 24.748 \][/tex]
### Step 3: Compute the Sample Variance
The sample variance is calculated by dividing this sum by the number of data points minus one. Here, our sample size \( n = 5 \).
[tex]\[ \text{Sample Variance} = \frac{24.748}{5 - 1} \][/tex]
[tex]\[ \text{Sample Variance} = \frac{24.748}{4} \][/tex]
[tex]\[ \text{Sample Variance} = 6.187 \][/tex]
### Step 4: Compute the Standard Deviation
The standard deviation is the square root of the sample variance.
[tex]\[ \text{Standard Deviation} = \sqrt{6.187} \][/tex]
[tex]\[ \text{Standard Deviation} \approx 2.487 \][/tex]
### Step 5: Round to Two Decimal Places
Finally, we round the sample variance and standard deviation to two decimal places.
[tex]\[ \text{Sample Variance} \approx 6.14 \][/tex]
[tex]\[ \text{Standard Deviation} \approx 2.48 \][/tex]
### Results
Variance \( \square \) = \( 6.14 \)
Standard Deviation [tex]\( \square \)[/tex] = [tex]\( 2.48 \)[/tex]
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