The Mean & Variance of the marks obtained by the student in a test are 10 and 4 respectively. Later, themarks of of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then theirnew variance is equal to :

Options

A. 3.92
B. 3.96
C. 4.04
D. 4.08

Answer

To solve the problem step by step, we will follow the given information and derive the new variance after the marks of one student are increased.

### Given:
- Initial Mean (μ) = 10
- Initial Variance (σ²) = 4
- Marks of one student increased from 8 to 12
- New Mean (μ') = 10.2

### Step 1: Determine the number of students (n)

Using the formula for the mean:
\[ \text{Mean} = \frac{\text{Sum of all marks}}{n} \]
Let the sum of the marks of the other students be \( S \). Then:
\[ \frac{S + 8}{n} = 10 \]
This implies:
\[ S + 8 = 10n \implies S = 10n - 8 \]

### Step 2: Calculate the initial variance

The formula for variance is given by:
\[ \text{Variance} = \frac{\sum (x_i^2)}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
Given that variance is 4, we have:
\[ 4 = \frac{\sum (x_i^2) + 8^2}{n} - 10^2 \]
Substituting \( 8^2 = 64 \) and \( 10^2 = 100 \):
\[ 4 = \frac{\sum (x_i^2) + 64}{n} - 100 \]
Rearranging gives:
\[ \frac{\sum (x_i^2) + 64}{n} = 104 \]
\[ \sum (x_i^2) + 64 = 104n \implies \sum (x_i^2) = 104n - 64 \]

### Step 3: Calculate the new mean after the increase

After increasing the student's marks from 8 to 12, the new sum of the marks becomes:
\[ S + 12 = 10n - 8 + 12 = 10n + 4 \]
The new mean is given by:
\[ \frac{10n + 4}{n} = 10.2 \]
Multiplying through by \( n \):
\[ 10n + 4 = 10.2n \]
Rearranging gives:
\[ 0.2n = 4 \implies n = 20 \]

### Step 4: Calculate the new variance

Now substituting \( n = 20 \) into the equation for \( \sum (x_i^2) \):
\[ \sum (x_i^2) = 104(20) - 64 = 2080 - 64 = 2016 \]
Now we calculate the new variance:
\[ \text{New Variance} = \frac{\sum (x_i^2) + 12^2}{n} - \left(\frac{10.2n}{n}\right)^2 \]
Substituting \( 12^2 = 144 \):
\[ \text{New Variance} = \frac{2016 + 144}{20} - (10.2)^2 \]
Calculating:
\[ \text{New Variance} = \frac{2160}{20} - 104.04 = 108 - 104.04 = 3.96 \]

### Final Answer:
The new variance is \( 3.96 \).