The distributions X and Y with total number of observations 36 and 64 and means 4 and 3 respectively are combined. What is the mean of the resulting distribution X+Y?

Options

A. 3.26
B. 3.32
C. 3.36
D. 3.42

Answer

To find the mean of the combined distributions X and Y, we can follow these steps:

### Step 1: Calculate the total value of distribution X
The total value of distribution X can be calculated using the formula:
\[ \text{Total value of X} = \text{Mean of X} \times \text{Number of observations in X} \]
Given:
- Mean of X = 4
- Number of observations in X = 36

Calculating:
\[ \text{Total value of X} = 4 \times 36 = 144 \]

### Step 2: Calculate the total value of distribution Y
Similarly, we calculate the total value of distribution Y:
\[ \text{Total value of Y} = \text{Mean of Y} \times \text{Number of observations in Y} \]
Given:
- Mean of Y = 3
- Number of observations in Y = 64

Calculating:
\[ \text{Total value of Y} = 3 \times 64 = 192 \]

### Step 3: Calculate the total value of the combined distribution X + Y
Now, we add the total values of distributions X and Y:
\[ \text{Total value of X + Y} = \text{Total value of X} + \text{Total value of Y} \]
Calculating:
\[ \text{Total value of X + Y} = 144 + 192 = 336 \]

### Step 4: Calculate the total number of observations in the combined distribution
The total number of observations in the combined distribution is:
\[ \text{Total observations} = \text{Number of observations in X} + \text{Number of observations in Y} \]
Calculating:
\[ \text{Total observations} = 36 + 64 = 100 \]

### Step 5: Calculate the mean of the combined distribution
Finally, we find the mean of the combined distribution using the formula:
\[ \text{Mean of X + Y} = \frac{\text{Total value of X + Y}}{\text{Total observations}} \]
Calculating:
\[ \text{Mean of X + Y} = \frac{336}{100} = 3.36 \]

### Final Answer
The mean of the resulting distribution X + Y is **3.36**.

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