The means and variance of n observations `x_(1),x_(2),x_(3),…x_(n)` are 0 and 5 respectively. If `sum_(i=1)^(n) x_(i)^(2) = 400`, then find the value of n

Answer

To solve the problem, we need to find the value of \( n \) given the mean and variance of \( n \) observations \( x_1, x_2, x_3, \ldots, x_n \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Mean (\( \mu \)) = 0
- Variance (\( \sigma^2 \)) = 5
- \( \sum_{i=1}^{n} x_i^2 = 400 \)

2. **Recall the Formula for Variance**:
The formula for variance is given by:
\[ \sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - \mu^2 \]

3. **Substitute the Known Values**:
Since the mean is 0, we can simplify the variance formula:
\[ 5 = \frac{400}{n} - 0^2 \]
Thus, we have:
\[ 5 = \frac{400}{n} \]

4. **Rearranging the Equation**:
To find \( n \), we can rearrange the equation:
\[ 5n = 400 \]

5. **Solve for \( n \)**:
Now, divide both sides by 5:
\[ n = \frac{400}{5} = 80 \]

### Final Answer:
The value of \( n \) is \( 80 \).

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