If for a sample size of `10`, `sum_(i=1)^(10)(x_i-5)^2=350 and sum_(i=1)^(10)(x_i-6)=20` , then the variance is

Options

A. 23
B. 24
C. 25
D. 26

Answer

To find the variance based on the given information, we will follow these steps:

### Step 1: Understand the given information
We have:
- Sample size \( n = 10 \)
- \( \sum_{i=1}^{10} (x_i - 5)^2 = 350 \)
- \( \sum_{i=1}^{10} (x_i - 6) = 20 \)

### Step 2: Calculate \( \sum_{i=1}^{10} x_i \)
From the second equation, we can express it as:
\[ \sum_{i=1}^{10} (x_i - 6) = \sum_{i=1}^{10} x_i - 60 = 20 \]
This leads to:
\[ \sum_{i=1}^{10} x_i = 20 + 60 = 80 \]

### Step 3: Calculate \( \sum_{i=1}^{10} (x_i - 5) \)
We can relate \( \sum_{i=1}^{10} (x_i - 5) \) to \( \sum_{i=1}^{10} x_i \):
\[ \sum_{i=1}^{10} (x_i - 5) = \sum_{i=1}^{10} x_i - 50 = 80 - 50 = 30 \]

### Step 4: Use the variance formula
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n} \]
Where \( \bar{x} \) is the mean. We can express it as:
\[ \sigma^2 = \frac{\sum_{i=1}^{10} (x_i - 5)^2}{10} - \left(\frac{\sum_{i=1}^{10} (x_i - 5)}{10}\right)^2 \]

### Step 5: Substitute the values
We know:
- \( \sum_{i=1}^{10} (x_i - 5)^2 = 350 \)
- \( \sum_{i=1}^{10} (x_i - 5) = 30 \)

Now substituting these values into the variance formula:
\[ \sigma^2 = \frac{350}{10} - \left(\frac{30}{10}\right)^2 \]
Calculating each term:
\[ \sigma^2 = 35 - 3^2 = 35 - 9 = 26 \]

### Final Answer
The variance is \( 26 \).

---