Two sample sizes of 50 and 100 are given . The mean of these samples respectively are 56 and 50. Find the mean of the size 150 by combining the two samples.

Answer

To find the combined mean of two samples with given sizes and means, we can follow these steps:

### Step 1: Identify the given data
- Sample A:
- Size (n₁) = 50
- Mean (Ā) = 56
- Sample B:
- Size (n₂) = 100
- Mean (B̄) = 50

### Step 2: Calculate the total sums for each sample
The total sum for each sample can be calculated using the formula:
\[ \text{Total Sum} = \text{Mean} \times \text{Size} \]

For Sample A:
\[ \text{Total Sum of A} = Ā \times n₁ = 56 \times 50 = 2800 \]

For Sample B:
\[ \text{Total Sum of B} = B̄ \times n₂ = 50 \times 100 = 5000 \]

### Step 3: Calculate the combined total sum
Now, we can find the combined total sum of both samples:
\[ \text{Combined Total Sum} = \text{Total Sum of A} + \text{Total Sum of B} \]
\[ \text{Combined Total Sum} = 2800 + 5000 = 7800 \]

### Step 4: Calculate the combined size
The combined size of the two samples is:
\[ \text{Combined Size} = n₁ + n₂ = 50 + 100 = 150 \]

### Step 5: Calculate the combined mean
The combined mean (C̄) can be calculated using the formula:
\[ C̄ = \frac{\text{Combined Total Sum}}{\text{Combined Size}} \]
\[ C̄ = \frac{7800}{150} = 52 \]

### Final Answer
The mean of the combined sample of size 150 is **52**.

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