For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is
(1) `(11)/2`
(2) 2
(3) `(13)/2`
(4) `5/2`

Options

A. `(13)/(2)`
B. `(5)/(2)`
C. `(11)/(2)`
D. 6

Explanation

To find the variance of the combined data set from two separate data sets, we can follow these steps:

### Step 1: Identify the given data
We have two data sets:
- Data set X:
- Size (n₁) = 5
- Variance (σ²₁) = 4
- Mean (μ₁) = 2
- Data set Y:
- Size (n₂) = 5
- Variance (σ²₂) = 5
- Mean (μ₂) = 4

### Step 2: Calculate the sum of squares for each data set
The formula for variance is:
\[ \sigma^2 = \frac{\Sigma x_i^2}{n} - \left(\frac{\Sigma x_i}{n}\right)^2 \]
From this, we can derive the sum of squares for each data set.

**For data set X:**
\[ \sigma^2_1 = \frac{\Sigma x_i^2}{n_1} - \mu_1^2 \]
Substituting the known values:
\[ 4 = \frac{\Sigma x_i^2}{5} - 2^2 \]
\[ 4 = \frac{\Sigma x_i^2}{5} - 4 \]
\[ \frac{\Sigma x_i^2}{5} = 8 \implies \Sigma x_i^2 = 40 \]

**For data set Y:**
\[ \sigma^2_2 = \frac{\Sigma y_i^2}{n_2} - \mu_2^2 \]
Substituting the known values:
\[ 5 = \frac{\Sigma y_i^2}{5} - 4^2 \]
\[ 5 = \frac{\Sigma y_i^2}{5} - 16 \]
\[ \frac{\Sigma y_i^2}{5} = 21 \implies \Sigma y_i^2 = 105 \]

### Step 3: Calculate the variance of the combined data set
The combined variance formula is given by:
\[ \sigma^2_z = \frac{1}{n_1 + n_2} \left(\Sigma x_i^2 + \Sigma y_i^2\right) - \left(\frac{n_1 \mu_1 + n_2 \mu_2}{n_1 + n_2}\right)^2 \]
Substituting the values:
\[ n_1 + n_2 = 5 + 5 = 10 \]
\[ \sigma^2_z = \frac{1}{10} \left(40 + 105\right) - \left(\frac{5 \cdot 2 + 5 \cdot 4}{10}\right)^2 \]
\[ = \frac{145}{10} - \left(\frac{10}{10}\right)^2 \]
\[ = \frac{145}{10} - 1 \]
\[ = \frac{145}{10} - \frac{10}{10} = \frac{135}{10} = \frac{11}{2} \]

### Conclusion
The variance of the combined data set is:
\[ \frac{11}{2} \]

### Answer
The correct option is (1) \(\frac{11}{2}\).