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 numbers and the resulting mean after multiplying each observation by 3, follow these steps:

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

### Step 2: Calculate the Sum of Observations
Add all the observations together:
\[ 8 + 12 + 16 + 22 + 10 + 4 \]
Calculating this step-by-step:
- \(8 + 12 = 20\)
- \(20 + 16 = 36\)
- \(36 + 22 = 58\)
- \(58 + 10 = 68\)
- \(68 + 4 = 72\)

So, the sum of the observations is \(72\).

### Step 3: Count the Number of Observations
Count the total number of observations:
There are 6 observations.

### Step 4: Calculate the Mean
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} = \frac{72}{6} \]
Calculating this gives:
\[ \text{Mean} = 12 \]

### Step 5: Calculate the Resulting Mean After Multiplication
Now, if each observation is multiplied by 3, the new mean can be calculated as:
\[ \text{Resulting Mean} = 3 \times \text{Original Mean} = 3 \times 12 \]
Calculating this gives:
\[ \text{Resulting Mean} = 36 \]

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

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