For two data sets, each with size 5, the variances are given to be 3 and 4 and the corresponding means are given 2 and 4, respectively. The variance of the combined data set is

Options

A. `(11)/(2)`
B. `(9)/(2)`
C. `(13)/(2)`
D. `(5)/(2)`

Answer

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

### Step 1: Understand the given data
We have two datasets:
- Dataset 1:
- Size (n1) = 5
- Variance (σ²₁) = 3
- Mean (μ₁) = 2
- Dataset 2:
- Size (n2) = 5
- Variance (σ²₂) = 4
- Mean (μ₂) = 4

### Step 2: Calculate the sum of squares for each dataset
The formula for variance is given by:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \mu^2 \]
From this, we can rearrange to find the sum of squares:
\[ \sum x_i^2 = n \cdot \sigma^2 + n \cdot \mu^2 \]

**For Dataset 1:**
\[ \sum x_i^2 = 5 \cdot 3 + 5 \cdot (2^2) = 15 + 20 = 35 \]

**For Dataset 2:**
\[ \sum y_i^2 = 5 \cdot 4 + 5 \cdot (4^2) = 20 + 80 = 100 \]

### Step 3: Calculate the combined mean
The combined mean (μ) of both datasets can be calculated as:
\[ \mu = \frac{n_1 \cdot \mu_1 + n_2 \cdot \mu_2}{n_1 + n_2} \]
Substituting the values:
\[ \mu = \frac{5 \cdot 2 + 5 \cdot 4}{5 + 5} = \frac{10 + 20}{10} = \frac{30}{10} = 3 \]

### Step 4: Calculate the variance of the combined dataset
The formula for the variance of the combined dataset is:
\[ \sigma^2 = \frac{\sum x_i^2 + \sum y_i^2}{n_1 + n_2} - \mu^2 \]
Substituting the values we calculated:
\[ \sigma^2 = \frac{35 + 100}{10} - 3^2 \]
\[ \sigma^2 = \frac{135}{10} - 9 = 13.5 - 9 = 4.5 \]

### Final Result
The variance of the combined dataset is **4.5**.

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