Let `x_(1), x_(2),…..,x_(3)` be n observations such that `Sigmax_(i)^(2)=300 and Sigmax_(1)=90`. Then a possible value of n among the following is

Options

A. 25
B. 18
C. 29
D. 22

Answer

To solve the problem, we need to find a possible value of \( n \) given the conditions \( \Sigma x_i^2 = 300 \) and \( \Sigma x_i = 90 \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
We have \( n \) observations \( x_1, x_2, \ldots, x_n \) such that:
\[ \Sigma x_i^2 = 300 \quad \text{(1)} \]
\[ \Sigma x_i = 90 \quad \text{(2)} \]

2. **Apply the Cauchy-Schwarz Inequality**:
The Cauchy-Schwarz inequality states that:
\[ \left( \Sigma x_i^2 \right) \left( n \right) \geq \left( \Sigma x_i \right)^2 \]
Substituting the values from (1) and (2):
\[ 300 \cdot n \geq 90^2 \]

3. **Calculate \( 90^2 \)**:
\[ 90^2 = 8100 \]
Thus, we can rewrite the inequality:
\[ 300n \geq 8100 \]

4. **Solve for \( n \)**:
Dividing both sides by 300:
\[ n \geq \frac{8100}{300} \]
Simplifying the right-hand side:
\[ n \geq 27 \]

5. **Determine Possible Values of \( n \)**:
Since \( n \) must be an integer, the smallest possible value of \( n \) is 28 (since \( n \) must be greater than 27).

### Conclusion:
The possible values of \( n \) must be greater than or equal to 28. You would need to check the options provided in the question to find which one is valid.