`If x_1, x_2, .....x_n` are n observations such that `sum_(i=1)^n (x_i)^2=400` and `sum_(i=1)^n x_i=100` then possible values of n among the following is

Options

A. 18
B. 20
C. 24
D. 27

Answer

To solve the problem, we need to analyze the given conditions about the observations \( x_1, x_2, \ldots, x_n \):

1. We know that:
\[ \sum_{i=1}^{n} (x_i)^2 = 400 \]
and
\[ \sum_{i=1}^{n} x_i = 100 \]

2. We will use the Cauchy-Schwarz inequality, which states that for any real numbers \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \):
\[ \left( \sum_{i=1}^{n} a_i b_i \right)^2 \leq \left( \sum_{i=1}^{n} a_i^2 \right) \left( \sum_{i=1}^{n} b_i^2 \right) \]

3. In our case, we can set \( a_i = 1 \) for all \( i \) (which means \( b_i = x_i \)):
\[ \left( \sum_{i=1}^{n} 1 \cdot x_i \right)^2 \leq \left( \sum_{i=1}^{n} 1^2 \right) \left( \sum_{i=1}^{n} (x_i)^2 \right) \]

4. This simplifies to:
\[ \left( \sum_{i=1}^{n} x_i \right)^2 \leq n \cdot \sum_{i=1}^{n} (x_i)^2 \]

5. Substituting the known sums:
\[ (100)^2 \leq n \cdot 400 \]

6. This gives us:
\[ 10000 \leq 400n \]

7. Dividing both sides by 400:
\[ n \geq \frac{10000}{400} = 25 \]

8. Therefore, the possible values of \( n \) must be greater than or equal to 25.

9. Now, we need to check which of the provided options for \( n \) is valid. The options are:
- 18 (not valid, as it is less than 25)
- 20 (not valid, as it is less than 25)
- 27 (valid, as it is greater than 25)

Thus, the possible value of \( n \) from the options given is **27**.