Let $x_{1},x_{2},...,x_{n}$ be 'n' observations such that $\Sigma x_{i}^{2}=400$ and $\Sigma x_{i}=80$. Then a possible value of 'n' among the following is:

Question

Accepted Answer

To determine the correct value of 'n' from the given options, we need to analyze the provided information about the observations $x_{1}, x_{2}, ..., x_{n}$.

We are given:
1. $\Sigma x_{i}^{2} = 400$
2. $\Sigma x_{i} = 80$

We need to find a possible value of 'n' such that these conditions hold true.

### Explanation:

1. **Mean and Variance:**
   - The mean of the observations is given by $\bar{x} = \frac{\Sigma x_{i}}{n} = \frac{80}{n}$.
   - The variance of the observations is given by $\text{Var}(x) = \frac{\Sigma x_{i}^{2}}{n} - \left(\frac{\Sigma x_{i}}{n}\right)^{2}$.

2. **Substitute the given values:**
   - Substitute $\Sigma x_{i}^{2} = 400$ and $\Sigma x_{i} = 80$ into the variance formula:
     $$
     \text{Var}(x) = \frac{400}{n} - \left(\frac{80}{n}\right)^{2} = \frac{400}{n} - \frac{6400}{n^2}
     $$

3. **Non-negative Variance:**
   - The variance must be non-negative, i.e., $\text{Var}(x) \geq 0$:
     $$
     \frac{400}{n} - \frac{6400}{n^2} \geq 0
     $$
   - Simplifying this inequality:
     $$
     400n^2 - 6400n \geq 0
     $$
     $$
     400n(n - 16) \geq 0
     $$

4. **Solve the inequality:**
   - The inequality $400n(n - 16) \geq 0$ implies $n(n - 16) \geq 0$.
   - This inequality holds for $n \geq 16$ or $n \leq 0$. Since $n$ must be a positive integer, we have $n \geq 16$.

5. **Choose the correct option:**
   - Among the given options (12, 9, 18, 15), the only value that satisfies $n \geq 16$ is 18.

### Conclusion:

- **Option 3 (18) is correct** because it satisfies the condition derived from the variance inequality, ensuring that the variance is non-negative.
- **Other Options:**
  - **Option 1 (12)** and **Option 4 (15)**: Both are less than 16, which does not satisfy the inequality $n \geq 16$.
  - **Option 2 (9)**: This is also less than 16 and does not satisfy the inequality.

Thus, based on the given conditions and logical reasoning, Option 3 (18) is the correct answer.