If `bar X_1 and bar X_2` are the means of two series such that `bar X_1 lt bar X_2 and bar X` is the mean of the combined series, then

Options

A. `bar(X) lt bar(X)_(1)`
B. `bar(X) gt bar(X)_(2)`
C. `bar(X)=(bar(X)_(1)+bar(X)_(2))/(2)`
D. `X_(1) lt bar(X) lt bar(X)_(2)`

Answer

To solve the problem step by step, we need to analyze the relationship between the means of the two series and the mean of the combined series.

### Step-by-Step Solution:

1. **Understanding the Means**:
Let \( \bar{X_1} \) be the mean of the first series and \( \bar{X_2} \) be the mean of the second series. We know that \( \bar{X_1} < \bar{X_2} \).

2. **Mean of Combined Series**:
The mean \( \bar{X} \) of the combined series can be expressed as:
\[ \bar{X} = \frac{\bar{X_1} + \bar{X_2}}{2} \]

3. **Establishing Inequalities**:
Since \( \bar{X_1} < \bar{X_2} \), we can add \( \bar{X_1} \) to both sides of the inequality:
\[ \bar{X_1} + \bar{X_1} < \bar{X_2} + \bar{X_1} \]
This simplifies to:
\[ 2\bar{X_1} < \bar{X_2} + \bar{X_1} \]

4. **Dividing by 2**:
Now, divide the entire inequality by 2:
\[ \bar{X_1} < \frac{\bar{X_2} + \bar{X_1}}{2} \]
This means:
\[ \bar{X_1} < \bar{X} \]

5. **Adding \( \bar{X_2} \)**:
Next, we can add \( \bar{X_2} \) to both sides of the original inequality \( \bar{X_1} < \bar{X_2} \):
\[ \bar{X_1} + \bar{X_2} < \bar{X_2} + \bar{X_2} \]
This simplifies to:
\[ \bar{X_1} + \bar{X_2} < 2\bar{X_2} \]

6. **Dividing by 2 Again**:
Dividing the entire inequality by 2 gives us:
\[ \frac{\bar{X_1} + \bar{X_2}}{2} < \bar{X_2} \]
This means:
\[ \bar{X} < \bar{X_2} \]

7. **Conclusion**:
From the above steps, we have established two inequalities:
\[ \bar{X_1} < \bar{X} < \bar{X_2} \]

### Final Result:
Thus, the final conclusion is:
\[ \bar{X_1} < \bar{X} < \bar{X_2} \]