If `overline (x_(1)) " and " overline (x_(2))` are the means of two distributions such that `overline (x_(1)) lt overline (x_(2)) " and " overline (x)` is the mean of the combined distriubtion, then

Options

A. `overline (x) lt overline (x_(1))`
B. `overline (x) gt overline (x_(2))`
C. `overline (x)=(overline (x_(1))+overline (x_(2)))/(2)`
D. `overline (x_(1)) lt overline (x) lt overline (x_(2))`

Explanation

To solve the problem step by step, we will analyze the means of two distributions and their combined mean.

### Step 1: Define the means and sample sizes
Let:
- \( \overline{x_1} \) = mean of the first distribution
- \( \overline{x_2} \) = mean of the second distribution
- \( n_1 \) = number of observations in the first distribution
- \( n_2 \) = number of observations in the second distribution

Given that \( \overline{x_1} < \overline{x_2} \).

### Step 2: Write the formula for the combined mean
The mean of the combined distribution \( \overline{x} \) can be calculated using the formula:
\[ \overline{x} = \frac{n_1 \overline{x_1} + n_2 \overline{x_2}}{n_1 + n_2} \]

### Step 3: Analyze the relationship between \( \overline{x} \), \( \overline{x_1} \), and \( \overline{x_2} \)
We will compare \( \overline{x} \) with \( \overline{x_1} \) and \( \overline{x_2} \).

#### Case 1: Compare \( \overline{x} \) with \( \overline{x_1} \)
We can rearrange the equation:
\[ \overline{x} - \overline{x_1} = \frac{n_1 \overline{x_1} + n_2 \overline{x_2}}{n_1 + n_2} - \overline{x_1} \]
Taking a common denominator:
\[ = \frac{n_1 \overline{x_1} + n_2 \overline{x_2} - (n_1 + n_2) \overline{x_1}}{n_1 + n_2} \]
\[ = \frac{n_2 \overline{x_2} - n_1 \overline{x_1}}{n_1 + n_2} \]

Since \( \overline{x_2} > \overline{x_1} \), it follows that \( n_2 \overline{x_2} - n_1 \overline{x_1} > 0 \) (assuming \( n_2 > 0 \)). Thus:
\[ \overline{x} > \overline{x_1} \]

#### Case 2: Compare \( \overline{x} \) with \( \overline{x_2} \)
Now, we compare \( \overline{x} \) with \( \overline{x_2} \):
\[ \overline{x} - \overline{x_2} = \frac{n_1 \overline{x_1} + n_2 \overline{x_2}}{n_1 + n_2} - \overline{x_2} \]
Taking a common denominator:
\[ = \frac{n_1 \overline{x_1} + n_2 \overline{x_2} - (n_1 + n_2) \overline{x_2}}{n_1 + n_2} \]
\[ = \frac{n_1 \overline{x_1} - n_2 \overline{x_2}}{n_1 + n_2} \]

Since \( \overline{x_2} > \overline{x_1} \), it follows that \( n_1 \overline{x_1} - n_2 \overline{x_2} < 0 \) (assuming \( n_1 > 0 \)). Thus:
\[ \overline{x} < \overline{x_2} \]

### Step 4: Combine the results
From the two cases, we have:
\[ \overline{x_1} < \overline{x} < \overline{x_2} \]

### Conclusion
Thus, the correct relationship among the means is:
\[ \overline{x_1} < \overline{x} < \overline{x_2} \]