Two data sets each of size 10 has the variance as 4 and k and the corresponding means as 2 and 4 respectively. If the variance of the combined data set is 5.5, then the value of k is equal to

Options

A. 5
B. 6
C. 4
D. 3

Answer

To solve the problem, we need to find the value of \( k \) given the variances and means of two data sets and the variance of the combined data set.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Data Set 1:
- Size (\( n_1 \)) = 10
- Variance (\( \sigma_1^2 \)) = 4
- Mean (\( \bar{x}_1 \)) = 2
- Data Set 2:
- Size (\( n_2 \)) = 10
- Variance (\( \sigma_2^2 \)) = \( k \)
- Mean (\( \bar{x}_2 \)) = 4
- Combined Variance = 5.5

2. **Calculate the Combined Mean:**
\[ \bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2} \]
Substituting the values:
\[ \bar{x} = \frac{10 \cdot 2 + 10 \cdot 4}{10 + 10} = \frac{20 + 40}{20} = \frac{60}{20} = 3 \]

3. **Calculate \( d_1 \) and \( d_2 \):**
- \( d_1 = \bar{x}_1 - \bar{x} = 2 - 3 = -1 \)
- \( d_2 = \bar{x}_2 - \bar{x} = 4 - 3 = 1 \)

4. **Use the Formula for Combined Variance:**
The formula for combined variance is given by:
\[ \sigma^2 = \frac{n_1 \sigma_1^2 + n_2 \sigma_2^2 + n_1 d_1^2 + n_2 d_2^2}{n_1 + n_2} \]
Substituting the known values:
\[ 5.5 = \frac{10 \cdot 4 + 10 \cdot k + 10 \cdot (-1)^2 + 10 \cdot 1^2}{10 + 10} \]

5. **Simplify the Equation:**
\[ 5.5 = \frac{40 + 10k + 10 + 10}{20} \]
\[ 5.5 = \frac{60 + 10k}{20} \]

6. **Multiply Both Sides by 20:**
\[ 5.5 \cdot 20 = 60 + 10k \]
\[ 110 = 60 + 10k \]

7. **Isolate \( k \):**
\[ 110 - 60 = 10k \]
\[ 50 = 10k \]
\[ k = \frac{50}{10} = 5 \]

### Final Answer:
The value of \( k \) is \( 5 \).