Find the mean of 8, 12, 16, 22, 10 and 4. Find the resulting mean, if each of the observations, given above, be :
multiplied by 3 and then divided by 2.

Answer

To solve the problem step by step, we will first find the mean of the original set of observations and then calculate the new mean after applying the transformations.

### Step 1: Calculate the Mean of the Original Observations

1. **List the Observations**: The given observations are 8, 12, 16, 22, 10, and 4.
2. **Sum the Observations**:
\[ \text{Sum} = 8 + 12 + 16 + 22 + 10 + 4 \]
Calculate the sum:
\[ \text{Sum} = 8 + 12 = 20 \]
\[ 20 + 16 = 36 \]
\[ 36 + 22 = 58 \]
\[ 58 + 10 = 68 \]
\[ 68 + 4 = 72 \]
So, the total sum is 72.

3. **Count the Number of Observations**: There are 6 observations.

4. **Calculate the Mean**:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} = \frac{72}{6} = 12 \]
Thus, the mean of the original observations is **12**.

### Step 2: Calculate the New Mean After Transformation

1. **Transformation**: Each observation is multiplied by 3 and then divided by 2. This can be expressed as:
\[ \text{New Mean} = \frac{3}{2} \times \text{Old Mean} \]

2. **Substitute the Old Mean**:
\[ \text{New Mean} = \frac{3}{2} \times 12 \]

3. **Calculate**:
\[ \frac{3 \times 12}{2} = \frac{36}{2} = 18 \]
Therefore, the resulting mean after the transformation is **18**.

### Final Answers:
- Mean of the original observations: **12**
- Mean after transformation: **18**

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