If `x_(1),x_(2),x_(3),.........,x_(n)` be n observations with mean `barx` and variance `sigma^(2)`. The mean and variance of `x_(1)+k,x_(2)+k,x_(3)+k,......,x_(n)+k` respectively are (i) `barx+k,sigma^(2)` (ii) `barx+k,sigma^(2)+k^(2)` (iii) `barx+k,(sigma+k)^(2)` (iv) `barx,sigma^(2)`

Options

A. `barx+k,sigma^(2)`
B. `barx+k,sigma^(2)+k^(2)`
C. `barx+k,(sigma+k)^(2)`
D. `barx,sigma^(2)`

Answer

To solve the problem, we need to find the mean and variance of the new set of observations \(x_1 + k, x_2 + k, x_3 + k, \ldots, x_n + k\), given that the original observations \(x_1, x_2, \ldots, x_n\) have a mean \(\bar{x}\) and variance \(\sigma^2\).

### Step-by-Step Solution:

1. **Identify the Mean of the Original Observations**:
The mean of the original observations is given by:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]
This implies that:
\[ x_1 + x_2 + x_3 + \ldots + x_n = n \bar{x} \]

2. **Calculate the Mean of the New Observations**:
The new observations are \(x_1 + k, x_2 + k, x_3 + k, \ldots, x_n + k\). The mean of these new observations can be calculated as follows:
\[ \text{Mean} = \frac{(x_1 + k) + (x_2 + k) + (x_3 + k) + \ldots + (x_n + k)}{n} \]
This simplifies to:
\[ \text{Mean} = \frac{(x_1 + x_2 + x_3 + \ldots + x_n) + nk}{n} = \frac{n \bar{x} + nk}{n} = \bar{x} + k \]

3. **Identify the Variance of the Original Observations**:
The variance of the original observations is given by:
\[ \sigma^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n} \]

4. **Calculate the Variance of the New Observations**:
The variance of the new observations \(x_1 + k, x_2 + k, \ldots, x_n + k\) is calculated as follows:
\[ \text{Variance} = \frac{((x_1 + k) - (\bar{x} + k))^2 + ((x_2 + k) - (\bar{x} + k))^2 + \ldots + ((x_n + k) - (\bar{x} + k))^2}{n} \]
This simplifies to:
\[ \text{Variance} = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n} = \sigma^2 \]

5. **Conclusion**:
From the calculations, we find that:
- The mean of the new observations is \(\bar{x} + k\).
- The variance of the new observations remains \(\sigma^2\).

Thus, the correct answer is:
**(i) \(\bar{x} + k, \sigma^2\)**.