If 150 is the mean of 200 observations and 100 is the mean of some 300 other observations, find the mean of the combination.

Answer

To find the mean of the combination of two sets of observations, we can follow these steps:

### Step 1: Identify the given values
- Let \( n_1 = 200 \) (the number of observations in the first set)
- Let \( x_1 = 150 \) (the mean of the first set)
- Let \( n_2 = 300 \) (the number of observations in the second set)
- Let \( x_2 = 100 \) (the mean of the second set)

### Step 2: Calculate the total sum of observations for each set
- The total sum of observations for the first set can be calculated as:
\[ \text{Sum}_1 = n_1 \times x_1 = 200 \times 150 = 30000 \]

- The total sum of observations for the second set can be calculated as:
\[ \text{Sum}_2 = n_2 \times x_2 = 300 \times 100 = 30000 \]

### Step 3: Calculate the total sum of all observations
- The total sum of all observations is:
\[ \text{Total Sum} = \text{Sum}_1 + \text{Sum}_2 = 30000 + 30000 = 60000 \]

### Step 4: Calculate the total number of observations
- The total number of observations is:
\[ \text{Total Observations} = n_1 + n_2 = 200 + 300 = 500 \]

### Step 5: Calculate the mean of the combined observations
- The mean of the combined observations is given by:
\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Total Observations}} = \frac{60000}{500} = 120 \]

### Final Answer
The mean of the combination of the two sets of observations is **120**.

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