Let `x_(1),x_(2),…x_(n)` be n observations such that `Sigmax_(i)^(2)=500 and Sigmax_(1)=100`. Then, an impossible value of n among the following is

Options

A. 24
B. 18
C. 29
D. 22

Answer

To solve the problem, we need to determine the impossible value of \( n \) given the conditions:

1. \( \sum x_i^2 = 500 \)
2. \( \sum x_i = 100 \)

### Step-by-Step Solution:

1. **Understanding the Variance Condition**:
The variance of a set of observations must be non-negative. The formula for variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{1}{n} \sum x_i^2 - \left( \frac{\sum x_i}{n} \right)^2 \]
This implies that:
\[ \frac{1}{n} \sum x_i^2 - \left( \frac{\sum x_i}{n} \right)^2 \geq 0 \]

2. **Substituting the Given Values**:
Substitute the given values into the variance formula:
\[ \frac{1}{n} \cdot 500 - \left( \frac{100}{n} \right)^2 \geq 0 \]

3. **Simplifying the Inequality**:
This simplifies to:
\[ \frac{500}{n} - \frac{10000}{n^2} \geq 0 \]
To eliminate the fractions, multiply through by \( n^2 \) (assuming \( n > 0 \)):
\[ 500n - 10000 \geq 0 \]

4. **Rearranging the Inequality**:
Rearranging gives:
\[ 500n \geq 10000 \]
Dividing both sides by 500:
\[ n \geq 20 \]

5. **Conclusion**:
The inequality \( n \geq 20 \) indicates that any value of \( n \) less than 20 is impossible. Therefore, the impossible value of \( n \) would be any integer less than 20.