Let `barx` be the mean of `x_(1), x_(2), ………, x_(n)` and `bary` be the mean of `y_(1), y_(2),……….,y_(n)`. If `barz` is the mean of `x_(1), x_(2), ……………..x_(n), y_(1), y_(2), …………,y_(n)`, then `barz` is equal to

Options

A. `(a+1/a)barx`
B. `(a + 1/a)barx/2`
C. `(a+1/a)barx/n`
D. `((a+1/a)barx)/(2n)`

Answer

To solve the problem, we need to find the mean \( \bar{z} \) of the combined data sets \( x_1, x_2, \ldots, x_n \) and \( y_1, y_2, \ldots, y_n \). We are given the means \( \bar{x} \) and \( \bar{y} \) of these data sets.

### Step-by-Step Solution:

1. **Understanding the means**:
- The mean \( \bar{x} \) of the data set \( x_1, x_2, \ldots, x_n \) is given by:
\[ \bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n} \]
- The mean \( \bar{y} \) of the data set \( y_1, y_2, \ldots, y_n \) is given by:
\[ \bar{y} = \frac{y_1 + y_2 + \ldots + y_n}{n} \]

2. **Expressing sums in terms of means**:
- From the definition of the mean, we can express the sums of the observations:
\[ x_1 + x_2 + \ldots + x_n = n \bar{x} \]
\[ y_1 + y_2 + \ldots + y_n = n \bar{y} \]

3. **Finding the total sum of all observations**:
- The total sum of all observations \( z_1, z_2, \ldots, z_{2n} \) (which includes both \( x \) and \( y \)) is:
\[ z_1 + z_2 + \ldots + z_{2n} = (x_1 + x_2 + \ldots + x_n) + (y_1 + y_2 + \ldots + y_n) \]
- Substituting the sums we found:
\[ z_1 + z_2 + \ldots + z_{2n} = n \bar{x} + n \bar{y} \]

4. **Calculating the mean \( \bar{z} \)**:
- The mean \( \bar{z} \) of the combined data set is given by:
\[ \bar{z} = \frac{z_1 + z_2 + \ldots + z_{2n}}{2n} \]
- Substituting the total sum:
\[ \bar{z} = \frac{n \bar{x} + n \bar{y}}{2n} \]

5. **Simplifying the expression**:
- We can simplify this expression:
\[ \bar{z} = \frac{n (\bar{x} + \bar{y})}{2n} = \frac{\bar{x} + \bar{y}}{2} \]

### Final Result:
Thus, the mean \( \bar{z} \) is equal to:
\[ \bar{z} = \frac{\bar{x} + \bar{y}}{2} \]