Let `x_(1), x_(2),….,x_(n)` be n observation such that `sum(x_(i))^(2)=400 and sumx_(i)=40`, then a possible value of n among the following is

Options

A. 5
B. 1
C. 2
D. 3

Answer

To solve the problem step by step, we will use the information given and apply the Cauchy-Schwarz inequality.

### Step 1: Understand the given information
We have two equations:
1. \(\sum_{i=1}^{n} x_i^2 = 400\)
2. \(\sum_{i=1}^{n} x_i = 40\)

### Step 2: Apply the Cauchy-Schwarz inequality
According to the Cauchy-Schwarz inequality, we have:
\[ \left( \sum_{i=1}^{n} x_i^2 \right) \left( \sum_{i=1}^{n} 1 \right) \geq \left( \sum_{i=1}^{n} x_i \right)^2 \]
Here, \(\sum_{i=1}^{n} 1 = n\) (since we have \(n\) observations).

### Step 3: Substitute the known values into the inequality
Substituting the known values into the inequality gives:
\[ \left( 400 \right) \left( n \right) \geq \left( 40 \right)^2 \]

### Step 4: Simplify the inequality
Calculating \(40^2\):
\[ 40^2 = 1600 \]
So the inequality becomes:
\[ 400n \geq 1600 \]

### Step 5: Solve for \(n\)
Dividing both sides by 400:
\[ n \geq \frac{1600}{400} = 4 \]

### Step 6: Determine possible values of \(n\)
Since \(n\) must be an integer greater than or equal to 4, the possible values of \(n\) can be 4, 5, 6, etc.

### Step 7: Check the options
From the options provided, we need to identify which values are greater than or equal to 4. The only value that satisfies \(n \geq 4\) from the options is 5.

### Final Answer
Thus, a possible value of \(n\) is **5**.

---