A data set of n observations has mean `2bar(x)` while another data set of 2n observations has mean `bar(x)` The mean of the combined data set of 3n observations will be equal to :

Options

A. `bar(x)`
B. `(3)/(2)bar(x)`
C. `(2)/(3)bar(x)`
D. `(4)/(3)bar(x)`

Answer

To solve the problem step by step, we will find the mean of the combined data set of 3n observations based on the given means of two separate data sets.

### Step 1: Understand the given data
We have two data sets:
1. The first data set has **n observations** with a mean of **2x̄**.
2. The second data set has **2n observations** with a mean of **x̄**.

### Step 2: Calculate the sum of the first data set
The mean of a data set is defined as the sum of the observations divided by the number of observations. Therefore, for the first data set:

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

This can be expressed as:

\[ 2x̄ = \frac{\Sigma_{i=1}^{n} x_i}{n} \]

Multiplying both sides by n gives us:

\[ \Sigma_{i=1}^{n} x_i = 2x̄ \cdot n \quad \text{(Equation 1)} \]

### Step 3: Calculate the sum of the second data set
Similarly, for the second data set with 2n observations:

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

This can be expressed as:

\[ x̄ = \frac{\Sigma_{i=1}^{2n} y_i}{2n} \]

Multiplying both sides by 2n gives us:

\[ \Sigma_{i=1}^{2n} y_i = x̄ \cdot 2n \quad \text{(Equation 2)} \]

### Step 4: Combine the sums of both data sets
Now, we need to find the mean of the combined data set of **3n observations**. The total sum of the observations from both data sets will be:

\[ \text{Total Sum} = \Sigma_{i=1}^{n} x_i + \Sigma_{i=1}^{2n} y_i \]

Substituting the values from Equation 1 and Equation 2:

\[ \text{Total Sum} = (2x̄ \cdot n) + (x̄ \cdot 2n) \]

### Step 5: Simplify the total sum
Combining the sums:

\[ \text{Total Sum} = 2x̄ n + 2x̄ n = 4x̄ n \]

### Step 6: Calculate the mean of the combined data set
The mean of the combined data set of 3n observations is given by:

\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Total Number of Observations}} = \frac{4x̄ n}{3n} \]

### Step 7: Simplify the mean
Cancelling n from the numerator and denominator:

\[ \text{Mean} = \frac{4x̄}{3} \]

### Final Answer
The mean of the combined data set of 3n observations is:

\[ \frac{4x̄}{3} \]