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.

Answer

To find the mean of the given observations and the resulting mean after multiplying each observation by 3, we can follow these steps:

### Step 1: List the Observations
The observations given are: 8, 12, 16, 22, 10, and 4.

### Step 2: Calculate the Sum of the Observations
We need to add all the observations together:
\[ 8 + 12 + 16 + 22 + 10 + 4 \]

### Step 3: Perform the Addition
Now, let's perform the addition step-by-step:
- First, add 8 and 12:
\[ 8 + 12 = 20 \]
- Next, add 16:
\[ 20 + 16 = 36 \]
- Then, add 22:
\[ 36 + 22 = 58 \]
- Now, add 10:
\[ 58 + 10 = 68 \]
- Finally, add 4:
\[ 68 + 4 = 72 \]

So, the total sum of the observations is 72.

### Step 4: Count the Number of Observations
The number of observations is 6.

### Step 5: Calculate the Mean
Now, we can calculate the mean using the formula:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} = \frac{72}{6} \]
Calculating this gives:
\[ \text{Mean} = 12 \]

### Step 6: Multiply Each Observation by 3
Next, we need to find the new mean if each observation is multiplied by 3. The new observations will be:
- \(8 \times 3 = 24\)
- \(12 \times 3 = 36\)
- \(16 \times 3 = 48\)
- \(22 \times 3 = 66\)
- \(10 \times 3 = 30\)
- \(4 \times 3 = 12\)

### Step 7: Calculate the New Mean
We can also find the new mean without calculating the new observations explicitly. The new mean can be calculated as:
\[ \text{New Mean} = 3 \times \text{Old Mean} = 3 \times 12 = 36 \]

### Final Answer
The mean of the original observations is 12, and the resulting mean after multiplying each observation by 3 is 36.

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