The variance of 10 observations is 16 and their mean is 12. If each observation is multiplied by 4, what is the new mean-

Options

A. (a) 12
B. (b) 16
C. c. 24
D. (d) 48.

Answer

To solve the problem step by step, we will follow the outlined approach based on the information provided.

### Step 1: Understand the given data
We know that:
- The variance of the observations is 16.
- The mean of the observations is 12.
- There are 10 observations.

### Step 2: Write the formula for the mean
The mean (\( \bar{x} \)) of the observations can be calculated using the formula:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_{10}}{10} \]
Given that the mean is 12, we can express this as:
\[ \frac{x_1 + x_2 + x_3 + \ldots + x_{10}}{10} = 12 \]

### Step 3: Calculate the sum of the observations
To find the sum of the observations, we can rearrange the mean formula:
\[ x_1 + x_2 + x_3 + \ldots + x_{10} = 12 \times 10 = 120 \]

### Step 4: Multiply each observation by 4
Now, we need to find the new mean when each observation is multiplied by 4. The new observations will be:
\[ 4x_1, 4x_2, 4x_3, \ldots, 4x_{10} \]

### Step 5: Write the formula for the new mean
The new mean (\( \bar{x}' \)) can be calculated as:
\[ \bar{x}' = \frac{4x_1 + 4x_2 + 4x_3 + \ldots + 4x_{10}}{10} \]

### Step 6: Factor out the 4
We can factor out the 4 from the numerator:
\[ \bar{x}' = \frac{4(x_1 + x_2 + x_3 + \ldots + x_{10})}{10} \]

### Step 7: Substitute the sum of the observations
We already calculated that the sum \( x_1 + x_2 + x_3 + \ldots + x_{10} = 120 \). Substituting this value into the equation gives:
\[ \bar{x}' = \frac{4 \times 120}{10} \]

### Step 8: Calculate the new mean
Now, we can calculate:
\[ \bar{x}' = \frac{480}{10} = 48 \]

### Final Answer
The new mean after multiplying each observation by 4 is **48**.

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