The mean and the variance of 10 observations are given to be 4 and 2 respectively. If every. observation is multiplied by 2 the mean and the variance of the new series will be respectively.

Options

A. 8and20
B. 8and4
C. 8and8
D. 80and40

Answer

To solve the problem step by step, we will calculate the new mean and variance after multiplying each observation by 2.

### Step 1: Understand the given information
We are given:
- Mean of 10 observations (original mean) = 4
- Variance of 10 observations (original variance) = 2
- Number of observations (n) = 10

### Step 2: Calculate the new mean
When every observation is multiplied by a constant (in this case, 2), the new mean can be calculated using the formula:
\[ \text{New Mean} = k \times \text{Original Mean} \]
where \( k \) is the constant by which each observation is multiplied.

Substituting the values:
\[ \text{New Mean} = 2 \times 4 = 8 \]

### Step 3: Calculate the new variance
The variance changes when each observation is multiplied by a constant. The new variance can be calculated using the formula:
\[ \text{New Variance} = k^2 \times \text{Original Variance} \]
where \( k \) is the constant.

Substituting the values:
\[ \text{New Variance} = 2^2 \times 2 = 4 \times 2 = 8 \]

### Final Result
Thus, the new mean and the new variance after multiplying each observation by 2 are:
- New Mean = 8
- New Variance = 8

### Summary
The new mean and variance of the new series will be respectively 8 and 8.

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