In an experiment with 9 observation on `x`, the following results are available `Sigmax^(2)=360 and Sigma x=34`. One observation that was 8, was found to be wrong and was replaced by the correct value 10, then the corrected variance is

Options

A. `(250)/(9)`
B. 28
C. `(240)/(9)`
D. 26

Answer

To find the corrected variance after replacing the incorrect observation, we can follow these steps:

### Step 1: Identify the given values
We are given:
- Number of observations, \( n = 9 \)
- \( \Sigma x^2 = 360 \)
- \( \Sigma x = 34 \)
- Incorrect observation = 8
- Correct observation = 10

### Step 2: Calculate the corrected sum of observations
To find the corrected sum of observations (\( \Sigma x_{\text{corrected}} \)):
\[ \Sigma x_{\text{corrected}} = \Sigma x - \text{incorrect observation} + \text{correct observation} \]
Substituting the values:
\[ \Sigma x_{\text{corrected}} = 34 - 8 + 10 = 36 \]

### Step 3: Calculate the corrected sum of squares
To find the corrected sum of squares (\( \Sigma x^2_{\text{corrected}} \)):
\[ \Sigma x^2_{\text{corrected}} = \Sigma x^2 - (\text{incorrect observation})^2 + (\text{correct observation})^2 \]
Substituting the values:
\[ \Sigma x^2_{\text{corrected}} = 360 - 8^2 + 10^2 = 360 - 64 + 100 = 396 \]

### Step 4: Calculate the mean of the corrected observations
The mean (\( \bar{x} \)) of the corrected observations is given by:
\[ \bar{x} = \frac{\Sigma x_{\text{corrected}}}{n} \]
Substituting the values:
\[ \bar{x} = \frac{36}{9} = 4 \]

### Step 5: Calculate the corrected variance
The variance (\( \sigma^2 \)) is calculated using the formula:
\[ \sigma^2 = \frac{\Sigma x^2_{\text{corrected}}}{n} - \bar{x}^2 \]
Substituting the values:
\[ \sigma^2 = \frac{396}{9} - 4^2 \]
Calculating \( \frac{396}{9} \):
\[ \frac{396}{9} = 44 \]
Now substituting this back into the variance formula:
\[ \sigma^2 = 44 - 16 = 28 \]

### Final Result
The corrected variance is \( 28 \).

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