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 will use the Cauchy-Schwarz inequality, which states that for any sequences of real numbers \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \):

\[ \left( \sum_{i=1}^{n} a_i^2 \right) \left( \sum_{i=1}^{n} b_i^2 \right) \geq \left( \sum_{i=1}^{n} a_i b_i \right)^2 \]

In our case, we will let \( a_i = x_i \) and \( b_i = 1 \) for \( i = 1, 2, \ldots, n \).

### Step 1: Set up the equations
We are given:
- \( \sum_{i=1}^{n} x_i^2 = 300 \)
- \( \sum_{i=1}^{n} x_i = 90 \)

### Step 2: Apply the Cauchy-Schwarz inequality
Using the Cauchy-Schwarz inequality, we have:

\[ \left( \sum_{i=1}^{n} x_i^2 \right) \left( \sum_{i=1}^{n} 1^2 \right) \geq \left( \sum_{i=1}^{n} x_i \right)^2 \]

This simplifies to:

\[ \left( \sum_{i=1}^{n} x_i^2 \right) n \geq \left( \sum_{i=1}^{n} x_i \right)^2 \]

Substituting the known values:

\[ 300n \geq 90^2 \]

### Step 3: Calculate \( 90^2 \)
Calculating \( 90^2 \):

\[ 90^2 = 8100 \]

### Step 4: Rearrange the inequality
Now we can rearrange the inequality:

\[ 300n \geq 8100 \]

Dividing both sides by 300:

\[ n \geq \frac{8100}{300} \]

### Step 5: Simplify the fraction
Calculating the right-hand side:

\[ \frac{8100}{300} = 27 \]

### Step 6: Conclusion
Thus, we find that:

\[ n \geq 27 \]

Now, we need to check the possible values of \( n \) given in the options. The options include values less than 27 (which are not valid) and values greater than or equal to 27.

Among the options, if 29 is provided, it is greater than 27 and thus a possible value of \( n \).

### Final Answer:
The possible value of \( n \) is **29**.

---