In an experiment with 15 observations of x the following results were available `sum x^2=2830 ,sum x=170` one observation that was 20 was found to be wrong and it was replaced by its correct value of 30. Then the corrected variance is

Options

A. `78.00`
B. `188.66`
C. `177.33`
D. `8.33`

Explanation

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

### Step 1: Calculate the corrected sum of observations (Σx)
Given:
- Original sum of observations (Σx) = 170
- Incorrect observation = 20
- Correct observation = 30

The corrected sum of observations can be calculated as:
\[ \text{Corrected } \Sigma x = \Sigma x - \text{incorrect observation} + \text{correct observation} \]
\[ \text{Corrected } \Sigma x = 170 - 20 + 30 = 180 \]

### Step 2: Calculate the corrected sum of squares of observations (Σx²)
Given:
- Original sum of squares of observations (Σx²) = 2830
- Square of the incorrect observation = \(20^2 = 400\)
- Square of the correct observation = \(30^2 = 900\)

The corrected sum of squares can be calculated as:
\[ \text{Corrected } \Sigma x^2 = \Sigma x^2 - \text{(incorrect observation)}^2 + \text{(correct observation)}^2 \]
\[ \text{Corrected } \Sigma x^2 = 2830 - 400 + 900 = 3330 \]

### Step 3: Calculate the corrected variance
The formula for variance (σ²) is:
\[ \sigma^2 = \frac{1}{n} \Sigma x^2 - \left(\frac{1}{n} \Sigma x\right)^2 \]
Where \(n\) is the number of observations (which is 15).

Substituting the corrected values:
\[ \sigma^2 = \frac{1}{15} \times 3330 - \left(\frac{1}{15} \times 180\right)^2 \]

Calculating each term:
1. Calculate \(\frac{1}{15} \times 3330\):
\[ \frac{1}{15} \times 3330 = 222 \]

2. Calculate \(\frac{1}{15} \times 180\):
\[ \frac{1}{15} \times 180 = 12 \]
Now, square this value:
\[ 12^2 = 144 \]

Putting it all together:
\[ \sigma^2 = 222 - 144 = 78 \]

### Final Answer
The corrected variance is \(78\).

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