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 solve the problem step by step, we need to find the corrected variance after replacing the wrong observation. Here’s how we can do it:

### Step 1: Understand the given data
We have:
- \( n = 9 \) (number of observations)
- \( \Sigma x^2 = 360 \) (sum of squares of observations)
- \( \Sigma x = 34 \) (sum of observations)
- One observation (8) was found to be wrong and replaced by 10.

### Step 2: Calculate the corrected sum of squares (\( \Sigma x'^2 \))
To find the corrected sum of squares, we need to subtract the square of the wrong observation and add the square of the correct observation:
\[ \Sigma x'^2 = \Sigma x^2 - (8^2) + (10^2) \]
Calculating this:
\[ \Sigma x'^2 = 360 - 64 + 100 = 396 \]

### Step 3: Calculate the corrected sum of observations (\( \Sigma x' \))
To find the corrected sum of observations, we subtract the wrong observation and add the correct observation:
\[ \Sigma x' = \Sigma x - 8 + 10 \]
Calculating this:
\[ \Sigma x' = 34 - 8 + 10 = 36 \]

### Step 4: Calculate the corrected variance (\( \sigma'^2 \))
The formula for variance is given by:
\[ \sigma'^2 = \frac{\Sigma x'^2}{n} - \left(\frac{\Sigma x'}{n}\right)^2 \]
Substituting the values we have:
\[ \sigma'^2 = \frac{396}{9} - \left(\frac{36}{9}\right)^2 \]
Calculating each term:
\[ \sigma'^2 = 44 - 4^2 = 44 - 16 = 28 \]

### Final Answer
The corrected variance is \( \sigma'^2 = 28 \).

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