If the standard deviation of n observation `x_(1), x_(2),…….,x_(n)` is 5 and for another set of n observation `y_(1), y_(2),………., y_(n)` is 4, then the standard deviation of n observation
`x_(1)-y_(1), x_(2)-y_(2),………….,x_(n)-y_(n)` is

Options

A. 1
B. `(sqrt5)/(2)`
C. 5
D. Data insufficient

Answer

To find the standard deviation of the observations \( z_i = x_i - y_i \) for \( i = 1, 2, \ldots, n \), we can use the properties of variance and standard deviation.

### Step-by-Step Solution:

1. **Understand the relationship between standard deviation and variance**:
The standard deviation (SD) is the square root of the variance (Var). Therefore, if we know the standard deviations of two sets of observations, we can find the variance of the difference of these observations.

2. **Given data**:
- Standard deviation of \( x_i \): \( \sigma_x = 5 \)
- Standard deviation of \( y_i \): \( \sigma_y = 4 \)

3. **Calculate the variances**:
- Variance of \( x_i \): \( \text{Var}(x) = \sigma_x^2 = 5^2 = 25 \)
- Variance of \( y_i \): \( \text{Var}(y) = \sigma_y^2 = 4^2 = 16 \)

4. **Use the formula for the variance of the difference of two independent random variables**:
\[ \text{Var}(z) = \text{Var}(x - y) = \text{Var}(x) + \text{Var}(y) \]
Since \( x_i \) and \( y_i \) are independent, we can add their variances.

5. **Calculate the variance of \( z_i \)**:
\[ \text{Var}(z) = \text{Var}(x) + \text{Var}(y) = 25 + 16 = 41 \]

6. **Find the standard deviation of \( z_i \)**:
\[ \sigma_z = \sqrt{\text{Var}(z)} = \sqrt{41} \]

### Final Answer:
The standard deviation of the observations \( z_i = x_i - y_i \) is \( \sqrt{41} \).

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