Let `x_(1), x_(2),…,x_(n)` be n observations such that `sum x_(i)^(2) = 400` and `sum x_(i) = 80`. Then a possible value of n among the following is

Options

A. 15
B. 18
C. 9
D. 12

Answer

To solve the problem step by step, we will use the given information about the observations \( x_1, x_2, \ldots, x_n \).

### Step 1: Write down the given information.
We have:
- \( \sum x_i^2 = 400 \)
- \( \sum x_i = 80 \)

### Step 2: Use the formula for variance.
The formula for variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
Substituting the given values into the formula, we get:
\[ \sigma^2 = \frac{400}{n} - \left(\frac{80}{n}\right)^2 \]

### Step 3: Simplify the expression.
Now, simplify the second term:
\[ \left(\frac{80}{n}\right)^2 = \frac{6400}{n^2} \]
So, we can rewrite the variance as:
\[ \sigma^2 = \frac{400}{n} - \frac{6400}{n^2} \]

### Step 4: Set the variance to be non-negative.
Since variance cannot be negative, we set up the inequality:
\[ \frac{400}{n} - \frac{6400}{n^2} \geq 0 \]

### Step 5: Solve the inequality.
To solve the inequality, we can rearrange it:
\[ \frac{400}{n} \geq \frac{6400}{n^2} \]
Multiplying both sides by \( n^2 \) (assuming \( n > 0 \)):
\[ 400n \geq 6400 \]
Dividing both sides by 400:
\[ n \geq \frac{6400}{400} \]
Calculating the right-hand side:
\[ n \geq 16 \]

### Step 6: Identify possible values of \( n \).
We need to find a possible value of \( n \) from the options given:
1. 15
2. 18
3. 9
4. 12

Since \( n \) must be greater than or equal to 16, the only option that satisfies this condition is:
- **18**

### Conclusion:
Thus, the possible value of \( n \) is **18**.

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